SOCR - User contributions [en]

Formulas

2010-05-14T02:13:55Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted \ gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n \ even\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Inverted Beta]:<math> X_1/X_2, \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Beta_Distribution.html Beta to Inverted Beta]:<math> \frac{X}{1-X} \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Arctangent]:<math> zero \ truncate \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Hypoexponential to Erlang]:<math> \vec \alpha=\alpha \ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Exponential to Hypoexponential]:<math> \sum X_i\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Erlang to Exponential]:<math> n=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Makeham to Gompertz]:<math> \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Doubly noncentral t to Noncentral t]:<math> \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Exponential to F]:<math> \alpha=1, X_1/X_2\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Noncentral F to F]:<math> \delta \to 0\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Exponential to Hyperexponential]:<math> Mixture\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Hyperexponential to Exponential]:<math> \vec \alpha=\alpha \ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html IDB to Exponential]:<math>\delta=\kappa \to 0, \alpha=1/ \gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Exponential to Rayleigh]:<math> X^2\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Weibull to Exponential]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Exponential to Weibull]:<math> X^{1/\beta}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Muth to Exponential]:<math> \alpha=1, \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Standard uniform to Gompertz]:<math> \frac{log[1-(log X)(log \kappa)/\delta]}{log \kappa}\ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard uniform to Exponential Power]:<math> [log(1-log(1-X))/\gamma]^{1/\kappa}\ </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Error to Laplace]:<math> a=0, b=\alpha/2, c=2\ </math>
* [http://socr.ucla.edu/htmls/dist/Error_Distribution.html Laplace to Error]:<math> \alpha_1=\alpha_2 \ </math>

<hr>

* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-14T02:01:33Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted \ gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n \ even\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Inverted Beta]:<math> X_1/X_2, \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Beta_Distribution.html Beta to Inverted Beta]:<math> \frac{X}{1-X} \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Arctangent]:<math> zero \ truncate \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Hypoexponential to Erlang]:<math> \vec \alpha=\alpha \ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Exponential to Hypoexponential]:<math> \sum X_i\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Erlang to Exponential]:<math> n=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Makeham to Gompertz]:<math> \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_t_Distribution.html Doubly noncentral t to Noncentral t]:<math> \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Exponential to F]:<math> \alpha=1, X_1/X_2\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Noncentral F to F]:<math> \delta \to 0\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Exponential to Hyperexponential]:<math> Mixture\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Hyperexponential to Exponential]:<math> \vec \alpha=\alpha \ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html IDB to Exponential]:<math>\delta=\kappa \to 0, \alpha=1\ \gamma \ </math>
* []:<math> \ </math>
* []:<math> \ </math>
* []:<math> \ </math>
* []:<math> \ </math>
* []:<math> \ </math>

<hr>

* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-14T01:49:23Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted \ gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n \ even\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Inverted Beta]:<math> X_1/X_2, \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Beta_Distribution.html Beta to Inverted Beta]:<math> \frac{X}{1-X} \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Arctangent]:<math> zero \ truncate \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Hypoexponential to Erlang]:<math> \vec \alpha=\alpha \ </math>

<hr>

* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-14T01:47:04Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted \ gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n \ even\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Inverted Beta]:<math> X_1/X_2, \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Beta_Distribution.html Beta to Inverted Beta]:<math> \frac{X}{1-X} \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Arctangent]:<math> zero truncate \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Hypoexponential to Erlang]:<math> \beta=n \ </math>

<hr>

* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-14T01:42:02Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted \ gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n \ even\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Inverted Beta]:<math> X_1/X_2 \alpha=1 \ </math>

<hr>

* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T20:52:37Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted \ gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n \ even\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math>
<hr>

* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T19:42:57Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n \ even\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math>
<hr>

* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T19:39:36Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n \ even\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math>
<hr>

* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T19:39:04Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n even\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math>
<hr>

* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T19:33:05Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T19:32:14Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
* [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> sqrt{X}\ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha \sum {X_i}}\ </math>
* [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T19:23:31Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
\infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T19:22:17Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to
infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(Z-\mu)^2/(\mu^2 X)\ </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T19:13:51Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
* [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \ </math>
* [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:42:25Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:41:02Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:38:44Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:38:21Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
<hr>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted gamma \ <math>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:35:00Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:34:42Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ <math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:34:16Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
<hr>
* [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ <math>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:30:14Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:29:40Z

Shelley zhouyuhao:

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum \frac{x-\mu}{\sigma}^2\ </math>
* [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p </math>
<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:25:36Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> \delta=0 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-05-06T18:25:05Z

Shelley zhouyuhao: /* Transformations */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> \delta=0 \</math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-29T04:59:32Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-29T04:43:03Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-29T04:40:37Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-29T04:38:38Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-29T04:35:21Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://mathworld.wolfram.com/WeibullDistribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T05:17:11Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T05:16:32Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\lappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T05:14:00Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\
\frac{2(b-x)}{(b-a)(b-m)}, m \le x
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T05:11:46Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi 0<\mu<2\pi) \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T05:10:02Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)} \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T05:08:08Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T05:05:58Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda k(\lambda k)^{k-1}}{[1+(\lambda x)^k]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda x}{(1+\lambda x)^{k+1}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T05:02:55Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda k(\lambda k)^{k-1}}{[1+(\lambda x)^k]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> f(x)=\frac{\lambda x}{(1+\lambda x)^k+1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T05:01:41Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda k(\lambda k)^{k-1}}{[1+(\lambda x)^k]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T04:59:47Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda k(\lambda k)^{k-1}}{[1+(\lambda x)^k]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\
\frac{n}{b-a}(\frac{b-x}{b-a})^{n-1}, m\le x
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T04:56:42Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda k(\lambda k)^{k-1}}{[1+(\lambda x)^k]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T04:56:11Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> f(x)=\frac{\lambda k(\lamda k)^{k-1}}{[1+(\lamda x)^k]^2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T04:54:43Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T04:53:32Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T04:50:46Z

Shelley zhouyuhao:

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T04:49:30Z

Shelley zhouyuhao: /* Probability Density Functions (PDFs) */

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{\gamma/2}(\frac{1}{2}\gamma)^k}{k!}\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1} x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-27T04:47:59Z

Shelley zhouyuhao:

==Probability Density Functions (PDFs)==

* [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] PDF: <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob |General Normal]] PDF: <math>f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}</math>
* [http://en.wikipedia.org/wiki/Chi-square_distribution Chi-Square] PDF: <math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma] PDF: <math>x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!</math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta] PDF: <math> \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T] PDF: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!</math>
* [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson]] PDF: <math>\frac{e^{-\lambda} \lambda^k}{k!}\!</math>
* [http://en.wikipedia.org/wiki/Chi_distribution Chi] PDF: <math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy] PDF: <math>\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}</math>
* [http://en.wikipedia.org/wiki/Exponential_distribution Exponential] PDF: <math>
\lambda e^{-\lambda x},\; x \ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\
\mbox{1 - p if k = 0,} \\
\mbox{0 otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math>
* [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>.
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math>
* [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math>
* [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math>
* [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx </math>
* [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math>
* [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math>
* [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math>
* [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math>
* [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>
* [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math>
* [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math>
* [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!</math>
* [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!</math>
* [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math>
* [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math>
* [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math>
* [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!</math>
* [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math>
* [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!</math>
* [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!</math>
* [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math>
* [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!</math>
* [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!</math>
* [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math>
* [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math>
* [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math>
* [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta
(\kappa^x-1)}{log(\kappa)}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t.html Doubly Noncentral t]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Hyperexponential.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math>
* [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\
1 - x, 0 \leq x<1 \end{cases} \!</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][e^{\gamma/2}}(\frac{1}{2}\gamma)^k}{k!}\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1} x>0 \!</math>
* [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value.html Extreme value]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Lomax.html Lomax]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/von-Mises.html von Mises]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Generalized-Pareto.html Generalized Pareto]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Triangular.html Triangular]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Kolmogorov-Smirnov.html Kolmogorov-Smirnov]: <math> \!</math>
* [http://socr.ucla.edu/htmls/dist/Exponential-power.html Exponential Power]: <math> \!</math>

==Transformations==
* [[AP_Statistics_Curriculum_2007_Normal_Prob | Standard Normal to General Normal Transformation]]: <math>\mu+\sigma\times X</math>
* [[AP_Statistics_Curriculum_2007_Normal_Prob | General Normal to Standard Normal Transformation]]: <math>X-\mu \over \sigma</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi Transformation]: <math>|\ X |</math>
* [http://en.wikipedia.org/wiki/Normal_distribution Standard Normal to Chi-Square Transformation]: <math>\sum_{k=1}^{\nu} X_k^2</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to General Normal Transformation]: <math>\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Exponential Transformation]: The special case of <math>{\Gamma}(k=1, \theta=1/\lambda)\,</math> is equivalent to exponential <math>Exp(\lambda)</math>.
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Beta Transformation]: <math>X_1 \over X_1 + X_2</math>.
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: <math>n\longrightarrow\infty</math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: <math>n=1 \ </math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: <math>a + \alpha\times X</math>
* [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: <math>a=0; \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Fisher's F to Student's T]: <math>\sqrt X </math>
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math>
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid)
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math>
* [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math>
* [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math>
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math>
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math>
* [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math>
* [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution
* [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math>
* [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math>
* [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math>
* [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math>
* [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math>
* [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math>
* [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math>
* [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}}

Formulas

2010-04-22T20:28:56Z

Shelley zhouyuhao:

Formulas

2010-04-22T20:27:11Z

Shelley zhouyuhao:

Formulas

2010-04-22T20:26:33Z

Shelley zhouyuhao: