SOCR - User contributions [en]

Formulas

2008-12-01T12:44:59Z

Jczhang: /* 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/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/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><br /> 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>

==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>
* [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>

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

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Formulas

2008-12-01T12:14:15Z

Jczhang: /* 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/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/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><br /> 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>

==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>
* [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>

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

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

Formulas

2008-12-01T10:55:48Z

Jczhang: /* 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/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/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><br /> 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>

==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>
* [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>

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

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

Formulas

2008-12-01T10:32:22Z

Jczhang: /* 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/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/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><br /> 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>

==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>
* [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>

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

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

Formulas

2008-11-25T09:52:38Z

Jczhang: /* 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/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/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^s}{H_{N,s}}</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>
* [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>

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

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

Formulas

2008-11-25T09:44:38Z

Jczhang: /* 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/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/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^s}{H_{N,s}}</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>
* [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>

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

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

Formulas

2008-11-25T09:43:21Z

Jczhang: /* 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/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/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>

==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>
* [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>

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

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

Formulas

2008-11-25T06:53:06Z

Jczhang: /* 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/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/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>

==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>
* [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>

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

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

Formulas

2008-11-25T05:34:52Z

Jczhang: /* Probability Density Functions (PDFs) */

Formulas

2008-11-18T10:34:46Z

Jczhang: /* Transformations */

Formulas

2008-11-18T10:32:54Z

Jczhang: /* Transformations */

Formulas

2008-11-18T10:16:22Z

Jczhang: /* Transformations */

Formulas

2008-11-12T10:17:15Z

Jczhang: /* Probability Density Functions (PDFs) */

Formulas

2008-10-28T08:24:25Z

Jczhang: /* Probability Density Functions (PDFs) */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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/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/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 Distribution] 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 Distribution] 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 Distribution] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic Distribution] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^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>
* [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>

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

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

Formulas

2008-10-28T08:11:07Z

Jczhang: /* Probability Density Functions (PDFs) */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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/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/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 Distribution] 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 Distribution] 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 Distribution] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math>
* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic Distribution] PDF:

==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>
* [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>

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

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

Formulas

2008-10-28T05:58:39Z

Jczhang: /* Probability Density Functions (PDFs) */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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/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/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>

==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>
* [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>

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

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

Formulas

2008-10-21T11:03:39Z

Jczhang: /* Transformations */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, p </math> for <math> k = 1 </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/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/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>

==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>
* [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>

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

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

Formulas

2008-10-21T11:02:47Z

Jczhang: /* Transformations */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, p </math> for <math> k = 1 </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/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/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>

==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>
* [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 Laplace to Exponential Transformation]: <math> x_1 - x_2 \ </math>

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

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

Formulas

2008-10-21T11:01:06Z

Jczhang: /* Transformations */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, p </math> for <math> k = 1 </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/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/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>

==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>
* [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 Laplace to Exponential Transformation]: <math> x_1 - x_2 \</math>

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

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

Formulas

2008-10-21T11:00:14Z

Jczhang: /* Transformations */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, p </math> for <math> k = 1 </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/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/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>

==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>
* [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 Laplace to Exponential Transformation]: <math> X_1 - X_2 \</math>

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

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

Formulas

2008-10-21T10:45:19Z

Jczhang: /* Transformations */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, p </math> for <math> k = 1 </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/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/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>

==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>
* [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>

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

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

Formulas

2008-10-21T10:42:10Z

Jczhang: /* Transformations */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, p </math> for <math> k = 1 </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/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/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>

==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> n = 1 </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>
* [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>

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

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

Formulas

2008-10-21T10:31:58Z

Jczhang: /* Transformations */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, p </math> for <math> k = 1 </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/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/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>

==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> n = 1 </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>
* [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>

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

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

Formulas

2008-10-21T10:08:18Z

Jczhang: /* Probability Density Functions (PDFs) */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, p </math> for <math> k = 1 </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/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/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>

==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> n = 1 </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>

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

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

Formulas

2008-10-21T10:07:51Z

Jczhang: /* Probability Density Functions (PDFs) */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, \\ p </math> for <math> k = 1 </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/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/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>

==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> n = 1 </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>

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

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

Formulas

2008-10-21T09:59:04Z

Jczhang: /* Probability Density Functions (PDFs) */

This [[Main_Page | SOCR Wiki]] page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc.

==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> q = (1 - p) </math> for <math> k = 0, p </math> for <math> k = 1 </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/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/Geometric_Distribution.html Geometric] PMF: <math> (1-p)^{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>

==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> n = 1 </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>

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

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Formulas

2008-10-21T09:15:13Z

Jczhang: /* Transformations */

Formulas

2008-10-21T09:04:38Z

Jczhang: /* Transformations */

Formulas

2008-10-21T08:35:47Z

Jczhang: /* Probability Density Functions (PDFs) */

Formulas

2008-10-21T08:16:31Z

Jczhang: /* Transformations */

Formulas

2008-10-21T07:43:30Z

Jczhang: /* Probability Density Functions (PDFs) */

Formulas

2008-10-21T06:41:00Z

Jczhang: /* Transformations */

Formulas

2008-10-21T06:08:58Z

Jczhang: /* Probability Density Functions (PDFs) */

Formulas

2008-10-21T06:07:53Z

Jczhang: /* Probability Density Functions (PDFs) */