SOCR - User contributions [en]

Formulas

2011-01-18T13:03:10Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </math>
* [http://socr.ucla.edu/htmls/dist/Modified-Power-Series_Distribution.html Modified Power Series distributon]: <math> P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)} </math> where <math> A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: <math> P(X=x)=\binom{n}{x}{p^{x} q^{n-x} \over (1-q^{n} )} </math> where <math> x=1,2,...,n </math>
* [http://socr.ucla.edu/htmls/dist/Basic-Lagrangian-distribution-of-the-first-kind.html Basic Lagrangian distribution of the first kind (BLD1)]: <math> P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } (g(z))^{x} \right]_{z=0} </math> where <math> g(z) </math> is pgf , <math> g(0) </math> is not 0
* [http://socr.ucla.edu/htmls/dist/General-Basic-Lagrangian-distribution-of-the-first-kind.html General Basic Lagrangian distribution of the first kind (GLD1)]: <math> P(X=0)=f(0) ,
P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} , x>0</math> Where f(z) and g(z) are pgf , <math>\left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} >0</math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-delta_Distribution.html Binomial-delta distribution]: <math> P(X=x)={n\over x}\binom{{mx}}{x-n}p^{x-n} q^{n+mx-x} </math> for <math>x\ge n</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-Poisson_Distribution.html Binomial-Poisson distribution]: <math> P(X=x)=e^{-M} {(Mq^{m} )^{x} \over x!} {}_{2} F{}_{0} [1-x,-mx;{p\over Mq} ] </math> , for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Binomial-negative-binomial_Distribution.html Binomial-negative-binomial distribution]: <math> P(X=x)={\Gamma (k+x)\over x!\Gamma (x)} Q^{-k} \left({Pq^{m} \over Q} \right)^{x} {}_{2} F_{1} [1-x,-mx;1-x-k;{-pQ\over qP} ] </math> for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-delta _Distribution.html Poisson-delta distribution]: <math> P(X=x)={n\over x} {e^{-\theta x} (\theta x)^{x-n} \over (x-n)} </math> for <math>x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-Poisson_Distribution.html Poisson-Poisson distribution(also called "Generalized Poisson distribution")]: <math> P(X=x)=M(M+\theta x)^{x-1} e^{-(M+\theta x)} /x! </math> for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-binomial_Distribution.html Poisson-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} npq^{n-1} {}_{2} F_{0} [1-x,1-n;{p\over \theta qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-negative-binomial_Distribution.html Poisson-negative-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} kPQ^{-k-1} {}_{2} F_{0} [1-x,1+k;{-P\over \theta Qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-delta_Distribution.html Negative-binomial-delta distribution]: <math> P(X=x)={n\over x} {\Gamma (kx+x-1)\over (x-n)!\Gamma (kx)} \left({P\over Q} \right)^{x-n} Q^{-kx} , x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-Poisson_Distribution.html Negative-binomial-Poisson distribution]: <math> P(X=x)={e^{-M} M^{x} \over x!} Q^{-kx} {}_{2} F_{0} [1-x,kx;-;{-P\over MQ} ] </math> , for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-binomial_Distribution.html Negative-binomial-binomial distribution]: <math> P(X=0)=q^{n} </math> , <math>P(X=x)=npq^{n-1} {\Gamma (kx+x-1)\over x!\Gamma (kx)} \left({P\over Q} \right)^{x-1} Q^{-kx} {}_{2} F_{1} [1-x,1-n;2-x-kx;{-pQ\over Pq} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-negative-binomial_Distribution.html Negative-binomial-negative-binomial distribution]: <math> P(X=x)=(Q')^{-M} \left({P'\over Q'Q^{k} } \right)^{x} {\Gamma (M+x)\over x!\Gamma (M)} {}_{2} F_{1} [1-x,kx;1-M-x;{PQ'\over P'Q} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Weight-binomial_Distribution.html Weight binomial distribution]: <math> P(X=x)=w(x)p_{x} /\sum _{x}^{}w(x)p_{x}</math>
* [http://socr.ucla.edu/htmls/dist/Positive-Poisson_Distribution.html Positive Poisson distribution (conditional Poisson distribution)]: <math> P(X=x)=(e^{\theta } -1)^{-1} \theta ^{x} /x! , x=1,2,......</math>
* [http://socr.ucla.edu/htmls/dist/Left-truncated-Poisson_Distribution.html Left-truncated Poisson distribution]: <math> P(X=x)={e^{-\theta } \theta ^{x} \over x!} \left[1-e^{-\theta } \sum _{j=0}^{r_{1} -1}{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...</math>
* [http://socr.ucla.edu/htmls/dist/Right-truncated-Poisson_Distribution.html Right-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=0}^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=0,1,...,r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-truncated-Poisson_Distribution.html Doubly-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=r_{1} }^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...,r_{2} , 0<r_{1} <r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Misrecorded-Poisson_Distribution.html Misrecorded Poisson distribution]: <math> P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1</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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \alpha =1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gaussian_Distribution.html Lévy to Gaussian]:<math> \alpha \to 2</math>
* [http://socr.ucla.edu/htmls/dist/Power-series_Distribution.html Modified Power Series to Power series]:<math> u(c)=c \ </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html BLD1 to Geometric]:<math> g(z)=1-p+pz \ </math> where<math>0<p<1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Borel-Tanner_Distribution.html BLD1 to Borel-Tanner]:<math> g(z)=e^{\lambda (z-1)} , 0<\lambda \ </math>
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html GLD1 to Binomial]:<math> g(z)=1 \ </math> and <math>f(z)=(q'+p'z)^{n} \ </math> where <math>q'=1-p' \ </math> , <math>0<p'<1 \ </math>, and n is positive integer.
* [http://socr.ucla.edu/htmls/dist/Negative-binomial_Distribution.html GLD1 to Negative binomial]:<math> g(z)=1 \ </math> and <math> f(z)=(q'+p'z)^{n} \ </math> where <math> q'=1+P \ </math> , <math> 0 , and <math> n=-k<0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Binomial-delta_Distribution.html GLD1 to Binomial-delta]: <math> g(z)=(q+pz)^{m} \ </math> , <math> f(z)=z^{n} \ </math> , <math> mp<1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Binomial-Poisson_Distribution.html GLD1 to Binomial-Poisson]:<math> : g(z)=(q+pz)^{m} \ </math> , <math> f(z)=e^{M(z-1)} \ </math> , <math> mp<1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Binomial-negative-binomial_Distribution.html GLD1 to Binomial-negative-binomial]:<math> g(z)=(q+pz)^{m} \ </math> , <math> f(z)=(Q-Pz)^{-k} \ </math> , <math> mp<1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-delta_Distribution.html GLD1 to Poisson-delta]: <math> g(z)=e^{\theta (z-1)} \ </math>, <math> f(z)=z^{n} \ </math>, <math> \theta <1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-Poisson_Distribution.html GLD1 to Poisson-Poisson]: <math> g(z)=e^{\theta (z-1)} , f(z)=e^{M(z-1)} , \theta <1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-binomial_Distribution.html GLD1 to Poisson-binomial]: <math> g(z)=e^{\theta (z-1)} , f(z)=(q+pz)^{n} , \theta <1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-negative-binomial_Distribution.html GLD1 to Poisson-negative-binomial]: <math> g(z)=e^{\theta (z-1)} , f(z)=(Q-Pz)^{-k} , \theta <1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-delta_Distribution.html GLD1 to Negative-binomial-delta]: <math> g(z)=(Q-Pz)^{-k} , f(z)=z^{n} , kP<1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-Poisson_Distribution.html GLD1 to Negative-binomial-Poisson]: <math> g(z)=(Q-Pz)^{-k} , f(z)=e^{M(z-1)} , kP<1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-binomial_Distribution.html GLD1 to Negative-binomial-binomial]: <math> g(z)=(Q-Pz)^{-k} , f(z)=(q+pz)^{n} , kP<1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-negative-binomial_Distribution.html GLD1 to Negative-binomial-negative-binomial]: <math> g(z)=(Q-Pz)^{-k} , f(z)=(Q'-P'z)^{-M} , kP<1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Chi-Square to Poisson]: <math> \left(1-F_{\chi _{2(x+1)}^{2} } (2t/\tau )\right)-\left(1-F_{\chi _{2x}^{2} } (2t/\tau )\right) \ </math> and <math> \lambda =t/\tau \ </math>
* [http://socr.ucla.edu/htmls/dist/Positive-Poisson_Distribution.html Left-truncated Poisson to Positive Poisson]: <math> r_{1} =1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Right-truncated-Poisson_Distribution.html Doubly-truncated Poisson to Right-truncated Poisson]: <math> r_{1} =0 \ </math>

<hr>

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

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

Formulas

2011-01-18T13:01:06Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </math>
* [http://socr.ucla.edu/htmls/dist/Modified-Power-Series_Distribution.html Modified Power Series distributon]: <math> P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)} </math> where <math> A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: <math> P(X=x)=\binom{n}{x}{p^{x} q^{n-x} \over (1-q^{n} )} </math> where <math> x=1,2,...,n </math>
* [http://socr.ucla.edu/htmls/dist/Basic-Lagrangian-distribution-of-the-first-kind.html Basic Lagrangian distribution of the first kind (BLD1)]: <math> P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } (g(z))^{x} \right]_{z=0} </math> where <math> g(z) </math> is pgf , <math> g(0) </math> is not 0
* [http://socr.ucla.edu/htmls/dist/General-Basic-Lagrangian-distribution-of-the-first-kind.html General Basic Lagrangian distribution of the first kind (GLD1)]: <math> P(X=0)=f(0) ,
P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} , x>0</math> Where f(z) and g(z) are pgf , <math>\left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} >0</math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-delta_Distribution.html Binomial-delta distribution]: <math> P(X=x)={n\over x}\binom{{mx}}{x-n}p^{x-n} q^{n+mx-x} </math> for <math>x\ge n</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-Poisson_Distribution.html Binomial-Poisson distribution]: <math> P(X=x)=e^{-M} {(Mq^{m} )^{x} \over x!} {}_{2} F{}_{0} [1-x,-mx;{p\over Mq} ] </math> , for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Binomial-negative-binomial_Distribution.html Binomial-negative-binomial distribution]: <math> P(X=x)={\Gamma (k+x)\over x!\Gamma (x)} Q^{-k} \left({Pq^{m} \over Q} \right)^{x} {}_{2} F_{1} [1-x,-mx;1-x-k;{-pQ\over qP} ] </math> for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-delta _Distribution.html Poisson-delta distribution]: <math> P(X=x)={n\over x} {e^{-\theta x} (\theta x)^{x-n} \over (x-n)} </math> for <math>x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-Poisson_Distribution.html Poisson-Poisson distribution(also called "Generalized Poisson distribution")]: <math> P(X=x)=M(M+\theta x)^{x-1} e^{-(M+\theta x)} /x! </math> for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-binomial_Distribution.html Poisson-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} npq^{n-1} {}_{2} F_{0} [1-x,1-n;{p\over \theta qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-negative-binomial_Distribution.html Poisson-negative-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} kPQ^{-k-1} {}_{2} F_{0} [1-x,1+k;{-P\over \theta Qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-delta_Distribution.html Negative-binomial-delta distribution]: <math> P(X=x)={n\over x} {\Gamma (kx+x-1)\over (x-n)!\Gamma (kx)} \left({P\over Q} \right)^{x-n} Q^{-kx} , x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-Poisson_Distribution.html Negative-binomial-Poisson distribution]: <math> P(X=x)={e^{-M} M^{x} \over x!} Q^{-kx} {}_{2} F_{0} [1-x,kx;-;{-P\over MQ} ] </math> , for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-binomial_Distribution.html Negative-binomial-binomial distribution]: <math> P(X=0)=q^{n} </math> , <math>P(X=x)=npq^{n-1} {\Gamma (kx+x-1)\over x!\Gamma (kx)} \left({P\over Q} \right)^{x-1} Q^{-kx} {}_{2} F_{1} [1-x,1-n;2-x-kx;{-pQ\over Pq} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-negative-binomial_Distribution.html Negative-binomial-negative-binomial distribution]: <math> P(X=x)=(Q')^{-M} \left({P'\over Q'Q^{k} } \right)^{x} {\Gamma (M+x)\over x!\Gamma (M)} {}_{2} F_{1} [1-x,kx;1-M-x;{PQ'\over P'Q} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Weight-binomial_Distribution.html Weight binomial distribution]: <math> P(X=x)=w(x)p_{x} /\sum _{x}^{}w(x)p_{x}</math>
* [http://socr.ucla.edu/htmls/dist/Positive-Poisson_Distribution.html Positive Poisson distribution (conditional Poisson distribution)]: <math> P(X=x)=(e^{\theta } -1)^{-1} \theta ^{x} /x! , x=1,2,......</math>
* [http://socr.ucla.edu/htmls/dist/Left-truncated-Poisson_Distribution.html Left-truncated Poisson distribution]: <math> P(X=x)={e^{-\theta } \theta ^{x} \over x!} \left[1-e^{-\theta } \sum _{j=0}^{r_{1} -1}{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...</math>
* [http://socr.ucla.edu/htmls/dist/Right-truncated-Poisson_Distribution.html Right-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=0}^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=0,1,...,r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-truncated-Poisson_Distribution.html Doubly-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=r_{1} }^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...,r_{2} , 0<r_{1} <r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Misrecorded-Poisson_Distribution.html Misrecorded Poisson distribution]: <math> P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1</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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>

<hr>

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

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

Formulas

2011-01-18T12:52:09Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </math>
* [http://socr.ucla.edu/htmls/dist/Modified-Power-Series_Distribution.html Modified Power Series distributon]: <math> P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)} </math> where <math> A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: <math> P(X=x)=\left\binom{n}{x}\right{p^{x} q^{n-x} \over (1-q^{n} )} </math> where <math> x=1,2,...,n </math>
* [http://socr.ucla.edu/htmls/dist/Basic-Lagrangian-distribution-of-the-first-kind.html Basic Lagrangian distribution of the first kind (BLD1)]: <math> P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } (g(z))^{x} \right]_{z=0} </math> where <math> g(z) </math> is pgf , <math> g(0) </math> is not 0
* [http://socr.ucla.edu/htmls/dist/General-Basic-Lagrangian-distribution-of-the-first-kind.html General Basic Lagrangian distribution of the first kind (GLD1)]: <math> P(X=0)=f(0) ,
P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} , x>0</math> Where f(z) and g(z) are pgf , <math>\left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} >0</math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-delta_Distribution.html Binomial-delta distribution]: <math> P(X=x)={n\over x}\binom{{mx}}{x-n}p^{x-n} q^{n+mx-x} </math> for <math>x\ge n</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-Poisson_Distribution.html Binomial-Poisson distribution]: <math> P(X=x)=e^{-M} {(Mq^{m} )^{x} \over x!} {}_{2} F{}_{0} [1-x,-mx;{p\over Mq} ] </math> , for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Binomial-negative-binomial_Distribution.html Binomial-negative-binomial distribution]: <math> P(X=x)={\Gamma (k+x)\over x!\Gamma (x)} Q^{-k} \left({Pq^{m} \over Q} \right)^{x} {}_{2} F_{1} [1-x,-mx;1-x-k;{-pQ\over qP} ] </math> for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-delta _Distribution.html Poisson-delta distribution]: <math> P(X=x)={n\over x} {e^{-\theta x} (\theta x)^{x-n} \over (x-n)} </math> for <math>x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-Poisson_Distribution.html Poisson-Poisson distribution(also called "Generalized Poisson distribution")]: <math> P(X=x)=M(M+\theta x)^{x-1} e^{-(M+\theta x)} /x! </math> for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-binomial_Distribution.html Poisson-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} npq^{n-1} {}_{2} F_{0} [1-x,1-n;{p\over \theta qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-negative-binomial_Distribution.html Poisson-negative-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} kPQ^{-k-1} {}_{2} F_{0} [1-x,1+k;{-P\over \theta Qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-delta_Distribution.html Negative-binomial-delta distribution]: <math> P(X=x)={n\over x} {\Gamma (kx+x-1)\over (x-n)!\Gamma (kx)} \left({P\over Q} \right)^{x-n} Q^{-kx} , x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-Poisson_Distribution.html Negative-binomial-Poisson distribution]: <math> P(X=x)={e^{-M} M^{x} \over x!} Q^{-kx} {}_{2} F_{0} [1-x,kx;-;{-P\over MQ} ] </math> , for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-binomial_Distribution.html Negative-binomial-binomial distribution]: <math> P(X=0)=q^{n} </math> , <math>P(X=x)=npq^{n-1} {\Gamma (kx+x-1)\over x!\Gamma (kx)} \left({P\over Q} \right)^{x-1} Q^{-kx} {}_{2} F_{1} [1-x,1-n;2-x-kx;{-pQ\over Pq} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-negative-binomial_Distribution.html Negative-binomial-negative-binomial distribution]: <math> P(X=x)=(Q')^{-M} \left({P'\over Q'Q^{k} } \right)^{x} {\Gamma (M+x)\over x!\Gamma (M)} {}_{2} F_{1} [1-x,kx;1-M-x;{PQ'\over P'Q} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Weight-binomial_Distribution.html Weight binomial distribution]: <math> P(X=x)=w(x)p_{x} /\sum _{x}^{}w(x)p_{x}</math>
* [http://socr.ucla.edu/htmls/dist/Positive-Poisson_Distribution.html Positive Poisson distribution (conditional Poisson distribution)]: <math> P(X=x)=(e^{\theta } -1)^{-1} \theta ^{x} /x! , x=1,2,......</math>
* [http://socr.ucla.edu/htmls/dist/Left-truncated-Poisson_Distribution.html Left-truncated Poisson distribution]: <math> P(X=x)={e^{-\theta } \theta ^{x} \over x!} \left[1-e^{-\theta } \sum _{j=0}^{r_{1} -1}{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...</math>
* [http://socr.ucla.edu/htmls/dist/Right-truncated-Poisson_Distribution.html Right-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=0}^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=0,1,...,r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-truncated-Poisson_Distribution.html Doubly-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=r_{1} }^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...,r_{2} , 0<r_{1} <r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Misrecorded-Poisson_Distribution.html Misrecorded Poisson distribution]: <math> P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1</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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>

<hr>

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

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

Formulas

2011-01-18T12:49:32Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </math>
* [http://socr.ucla.edu/htmls/dist/Modified-Power-Series_Distribution.html Modified Power Series distributon]: <math> P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)} </math> where <math> A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: <math> P(X=x)=\left\binom{n}{x}\right{p^{x} q^{n-x} \over (1-q^{n} )} </math> where <math> x=1,2,...,n </math>
* [http://socr.ucla.edu/htmls/dist/Basic-Lagrangian-distribution-of-the-first-kind.html Basic Lagrangian distribution of the first kind (BLD1)]: <math> P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } (g(z))^{x} \right]_{z=0} </math> where <math> g(z) </math> is pgf , <math> g(0) </math> is not 0
* [http://socr.ucla.edu/htmls/dist/General-Basic-Lagrangian-distribution-of-the-first-kind.html General Basic Lagrangian distribution of the first kind (GLD1)]: <math> {P(X=0)=f(0)} \cr ,
P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} , x>0</math> Where f(z) and g(z) are pgf , <math>\left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} >0</math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-delta_Distribution.html Binomial-delta distribution]: <math> P(X=x)={n\over x}\binom{{mx}}{x-n}p^{x-n} q^{n+mx-x} </math> for <math>x\ge n</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-Poisson_Distribution.html Binomial-Poisson distribution]: <math> P(X=x)=e^{-M} {(Mq^{m} )^{x} \over x!} {}_{2} F{}_{0} [1-x,-mx;{p\over Mq} ] </math> , for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Binomial-negative-binomial_Distribution.html Binomial-negative-binomial distribution]: <math> P(X=x)={\Gamma (k+x)\over x!\Gamma (x)} Q^{-k} \left({Pq^{m} \over Q} \right)^{x} {}_{2} F_{1} [1-x,-mx;1-x-k;{-pQ\over qP} ] </math> for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-delta _Distribution.html Poisson-delta distribution]: <math> P(X=x)={n\over x} {e^{-\theta x} (\theta x)^{x-n} \over (x-n)} </math> for <math>x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-Poisson_Distribution.html Poisson-Poisson distribution(also called "Generalized Poisson distribution")]: <math> P(X=x)=M(M+\theta x)^{x-1} e^{-(M+\theta x)} /x! </math> for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-binomial_Distribution.html Poisson-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} npq^{n-1} {}_{2} F_{0} [1-x,1-n;{p\over \theta qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-negative-binomial_Distribution.html Poisson-negative-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} kPQ^{-k-1} {}_{2} F_{0} [1-x,1+k;{-P\over \theta Qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-delta_Distribution.html Negative-binomial-delta distribution]: <math> P(X=x)={n\over x} {\Gamma (kx+x-1)\over (x-n)!\Gamma (kx)} \left({P\over Q} \right)^{x-n} Q^{-kx} , x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-Poisson_Distribution.html Negative-binomial-Poisson distribution]: <math> P(X=x)={e^{-M} M^{x} \over x!} Q^{-kx} {}_{2} F_{0} [1-x,kx;-;{-P\over MQ} ] </math> , for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-binomial_Distribution.html Negative-binomial-binomial distribution]: <math> P(X=0)=q^{n} </math> , <math>P(X=x)=npq^{n-1} {\Gamma (kx+x-1)\over x!\Gamma (kx)} \left({P\over Q} \right)^{x-1} Q^{-kx} {}_{2} F_{1} [1-x,1-n;2-x-kx;{-pQ\over Pq} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-negative-binomial_Distribution.html Negative-binomial-negative-binomial distribution]: <math> P(X=x)=(Q')^{-M} \left({P'\over Q'Q^{k} } \right)^{x} {\Gamma (M+x)\over x!\Gamma (M)} {}_{2} F_{1} [1-x,kx;1-M-x;{PQ'\over P'Q} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Weight-binomial_Distribution.html Weight binomial distribution]: <math> P(X=x)=w(x)p_{x} /\sum _{x}^{}w(x)p_{x}</math>
* [http://socr.ucla.edu/htmls/dist/Positive-Poisson_Distribution.html Positive Poisson distribution (conditional Poisson distribution)]: <math> P(X=x)=(e^{\theta } -1)^{-1} \theta ^{x} /x! , x=1,2,......</math>
* [http://socr.ucla.edu/htmls/dist/Left-truncated-Poisson_Distribution.html Left-truncated Poisson distribution]: <math> P(X=x)={e^{-\theta } \theta ^{x} \over x!} \left[1-e^{-\theta } \sum _{j=0}^{r_{1} -1}{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...</math>
* [http://socr.ucla.edu/htmls/dist/Right-truncated-Poisson_Distribution.html Right-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=0}^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=0,1,...,r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-truncated-Poisson_Distribution.html Doubly-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=r_{1} }^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...,r_{2} , 0<r_{1} <r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Misrecorded-Poisson_Distribution.html Misrecorded Poisson distribution]: <math> P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1</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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>

<hr>

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

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

Formulas

2011-01-18T12:46:07Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </math>
* [http://socr.ucla.edu/htmls/dist/Modified-Power-Series_Distribution.html Modified Power Series distributon]: <math> P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)} </math> where <math> A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: <math> P(X=x)=\left\binom{n}{x}\right{p^{x} q^{n-x} \over (1-q^{n} )} </math> where <math> x=1,2,...,n </math>
* [http://socr.ucla.edu/htmls/dist/Basic-Lagrangian-distribution-of-the-first-kind.html Basic Lagrangian distribution of the first kind (BLD1)]: <math> P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } (g(z))^{x} \right]_{z=0} </math> where <math> g(z) </math> is pgf , <math> g(0) </math> is not 0
* [http://socr.ucla.edu/htmls/dist/General-Basic-Lagrangian-distribution-of-the-first-kind.html General Basic Lagrangian distribution of the first kind (GLD1)]: <math> {P(X=0)=f(0)} \cr ,
P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} , x>0</math> Where f(z) and g(z) are pgf , <math>\left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} >0</math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-delta_Distribution.html Binomial-delta distribution]: <math> P(X=x)={n\over x}\binom{{mx}}{x-n}p^{x-n} q^{n+mx-x} </math> for <math>x\ge n</math>
* [http://socr.ucla.edu/htmls/dist/Binomial-Poisson_Distribution.html Binomial-Poisson distribution]: <math> P(X=x)=e^{-M} {(Mq^{m} )^{x} \over x!} {}_{2} F{}_{0} [1-x,-mx;{p\over Mq} ] </math> , for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Binomial-negative-binomial_Distribution.html Binomial-negative-binomial distribution]: <math> P(X=x)={\Gamma (k+x)\over x!\Gamma (x)} Q^{-k} \left({Pq^{m} \over Q} \right)^{x} {}_{2} F_{1} [1-x,-mx;1-x-k;{-pQ\over qP} ] </math> for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-delta _Distribution.html Poisson-delta distribution]: <math> P(X=x)={n\over x} {e^{-\theta x} (\theta x)^{x-n} \over (x-n)} </math> for <math>x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-Poisson_Distribution.html Poisson-Poisson distribution(also called "Generalized Poisson distribution")]: <math> P(X=x)=M(M+\theta x)^{x-1} e^{-(M+\theta x)} /x! </math> for <math>x\ge 0 </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-binomial_Distribution.html Poisson-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} npq^{n-1} {}_{2} F_{0} [1-x,1-n;{p\over \theta qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Poisson-negative-binomial_Distribution.html Poisson-negative-binomial distribution]: <math> P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} kPQ^{-k-1} {}_{2} F_{0} [1-x,1+k;{-P\over \theta Qx} ] , x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-delta_Distribution.html Negative-binomial-delta distribution]: <math> P(X=x)={n\over x} {\Gamma (kx+x-1)\over (x-n)!\Gamma (kx)} \left({P\over Q} \right)^{x-n} Q^{-kx} , x\ge n </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-Poisson_Distribution.html Negative-binomial-Poisson distribution]: <math> P(X=x)={e^{-M} M^{x} \over x!} Q^{-kx} {}_{2} F_{0} [1-x,kx;-;{-P\over MQ} ] </math> , for <math>x\ge 0</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-binomial_Distribution.html Negative-binomial-binomial distribution]: <math> P(X=0)=q^{n} </math> , <math>P(X=x)=npq^{n-1} {\Gamma (kx+x-1)\over x!\Gamma (kx)} \left({P\over Q} \right)^{x-1} Q^{-kx} {}_{2} F_{1} [1-x,1-n;2-x-kx;{-pQ\over Pq} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial-negative-binomial_Distribution.html Negative-binomial-negative-binomial distribution]: <math> P(X=x)=(Q')^{-M} \left({P'\over Q'Q^{k} } \right)^{x} {\Gamma (M+x)\over x!\Gamma (M)} {}_{2} F_{1} [1-x,kx;1-M-x;{PQ'\over P'Q} ] </math> for <math>x\ge 1</math>
* [http://socr.ucla.edu/htmls/dist/Weight-binomial_Distribution.html Weight binomial distribution]: <math> P(X=x)=w(x)p_{x} /\sum _{x}^{}w(x)p_{x}</math>
* [http://socr.ucla.edu/htmls/dist/Positive-Poisson_Distribution.html Positive Poisson distribution (conditional Poisson distribution)]: <math> P(X=x)=(e^{\theta } -1)^{-1} \theta ^{x} /x! , x=1,2,......</math>
* [http://socr.ucla.edu/htmls/dist/Left-truncated-Poisson_Distribution.html Left-truncated Poisson distribution]: <math> P(X=x)={e^{-\theta } \theta ^{x} \over x!} \left[1-e^{-\theta } \sum _{j=0}^{r_{1} -1}{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...</math>
* [http://socr.ucla.edu/htmls/dist/Right-truncated-Poisson_Distribution.html Right-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=0}^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=0,1,...,r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Doubly-truncated-Poisson_Distribution.html Doubly-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=r_{1} }^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...,r_{2} , 0<r_{1} <r_{2}</math>
* [http://socr.ucla.edu/htmls/dist/Misrecorded-Poisson_Distribution.html Misrecorded Poisson distribution]: <math> P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1</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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:56:39Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:56:15Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Poisson-Poisson_Distribution.html GLD1 to Poisson-Poisson]: <math> g(z)=e^{\theta (z-1)} , f(z)=e^{M(z-1)} , \theta <1 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:45:04Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:44:31Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial_Distribution.html GLD1 to Negative binomial]:<math> g(z)=1 \ </math> and <math> f(z)=(q'+p'z)^{n} \ </math> where <math> q'=1+P \ </math> , <math> 0 , and <math> n=-k<0 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:43:48Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial_Distribution.html GLD1 to Negative binomial]:<math> g(z)=1 \ </math> and <math> f(z)=(q'+p'z)^{n} \ </math> where <math> q'=1+P \ </math> , </math> 0 , and </math> n=-k<0 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:41:57Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:41:25Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial_Distribution.html GLD1 to Negative binomial]:<math> g(z)=1 and f(z)=(q'+p'z)^{n} where q'=1+P,0<P, and n=-k<0 </math>

<hr>

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

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

Formulas

2011-01-11T13:40:28Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Negative-binomial_Distribution.html GLD1 to Negative binomial]:<math> g(z)=1$ and $f(z)=(q'+p'z)^{n} $ where $q'=1+P$,$0<P$, and $n=-k<0 </math>

<hr>

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

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

Formulas

2011-01-11T13:33:47Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:33:04Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \alpha =1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gaussian_Distribution.html Lévy to Gaussian]:<math> \alpha \to 2</math>
* [http://socr.ucla.edu/htmls/dist/Power-series_Distribution.html Modified Power Series to Power series]:<math> u(c)=c \ </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html BLD1 to Geometric]:<math> g(z)=1-p+pz \ </math> where <math>0<p<1 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:32:11Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \alpha =1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gaussian_Distribution.html Lévy to Gaussian]:<math> \alpha \to 2</math>
* [http://socr.ucla.edu/htmls/dist/Power-series_Distribution.html Modified Power Series to Power series]:<math> u(c)=c \ </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html BLD1 to Geometric]:<math> g(z)=1-p+pz \</math> where<math>0<p<1 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:31:23Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \alpha =1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gaussian_Distribution.html Lévy to Gaussian]:<math> \alpha \to 2</math>
* [http://socr.ucla.edu/htmls/dist/Power-series_Distribution.html Modified Power Series to Power series]:<math> u(c)=c \ </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html BLD1 to Geometric]:<math> g(z)=1-p+pz \<math> where</math>0<p<1 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:30:26Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \alpha =1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gaussian_Distribution.html Lévy to Gaussian]:<math> \alpha \to 2</math>
* [http://socr.ucla.edu/htmls/dist/Power-series_Distribution.html Modified Power Series to Power series]:<math> u(c)=c \ </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html BLD1 to Geometric]:<math> g(z)=1-p+pz$ where $0<p<1 \ </math>

<hr>

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

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

Formulas

2011-01-11T13:29:37Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \alpha =1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gaussian_Distribution.html Lévy to Gaussian]:<math> \alpha \to 2</math>
* [http://socr.ucla.edu/htmls/dist/Power-series_Distribution.html Modified Power Series to Power series]:<math> u(c)=c </math>
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html BLD1 to Geometric]:<math> g(z)=1-p+pz$ where $0<p<1 </math>

<hr>

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

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

Formulas

2011-01-11T13:26:00Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \alpha =1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Gaussian_Distribution.html Lévy to Gaussian]:<math> \alpha \to 2</math>

<hr>

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

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

Formulas

2011-01-11T13:24:34Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \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

2011-01-11T13:23:22Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \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

2011-01-11T13:22:03Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \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

2011-01-11T12:29:34Z

ChinPangHo: /* 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>
* [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 </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>
* [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math>
* [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math>

* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math>
* [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math>
* [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math>
* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math>
* [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math>

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

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