Formulas

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Probability Density Functions (PDFs)

Standard Normal PDF: $f(x)= {e^{-x^2} \over \sqrt{2 \pi}}$
General Normal PDF: $f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}$
Chi-Square PDF: $\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,$
Gamma PDF: $x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!$
Beta PDF: $\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!$
Student's T PDF: $\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!$
Poisson PDF: $\frac{e^{-\lambda} \lambda^k}{k!}\!$
Chi PDF: $\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}$
Cauchy PDF: $\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}$
Exponential PDF: $\lambda e^{-\lambda x},\; x \ge 0$
F Distribution PDF: $\frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) }$
Bernoulli PMF: $f(k;p) \begin{cases} \mbox{p if k = 1,} \\ \mbox{1 - p if k = 0,} \\ \mbox{0 otherwise} \end{cases}$
Binomial PMF: $\begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}$
Multinomial PMF: $f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}$ , where $x_1+x_2+\cdots+x_k=n$ , $p_1+p_2+\cdots+p_k=1$ , and $0 \le x_i \le n, 0 \le p_i \le 1$ .
Negative Binomial PMF: $\begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k$
Negative-Multinomial Binomial PMF: $P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}}$
Geometric PMF: $\begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p$
Erlang PDF: $\frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!}$
Laplace PDF: $\frac {1}{2b} \exp (- \frac{|x-\mu|}{b})$
Continuous Uniform PDF: $f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases}$
Discrete Uniform PMF: $f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases}$
Logarithmic PDF: $f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k}$
Logistic PDF: $f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2}$
Logistic-Exponential PDF: $f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0$
Power Function PDF: $f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha}$
Benford's Law: $P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d})$
Pareto PDF: $\frac {kx^k_m} {x^{k+1}}$
Non-Central Student T PDF: $f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx$
ArcSine PDF: $f(x) = \frac{1}{\pi \sqrt{x(1-x)}}$
Circle PDF: $f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r]$
U-Quadratic PDF: $\alpha \left ( x - \beta \right )^2$
Standard Uniform PDF: $U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases}$
Zipf: $\frac{1/(k+q)^s}{H_{N,s}}$
Inverse Gamma: $\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)$
Fisher-Tippett: $\frac{z\,e^{-z}}{\beta}\!$
where $z = e^{-\frac{x-\mu}{\beta}}\!$
Gumbel: $f(x) = e^{-x} e^{-e^{-x}}.$
HyperGeometric: ${{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}$
Log-Normal: $\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]$
Gilbrats: $\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]$
Hyperbolic Secant: $\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!$
Gompertz: $b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]$
Standard Cauchy: $f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!$
Rectangular: $f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!$
Beta-Binomial: $f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!$
Negative Hypergeometric: $f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!$
Standard Power: $f(x; \beta) = \beta x^{\beta - 1} \!$
Power_Series: $f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!$
Zeta: $f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!$
Logarithm: $f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!$
Beta_Pascal: $f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!$
Gamma_Poisson: $f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!$
Pascal: $f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!$
Polya: $f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!$
Normal-Gamma: $f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!$
Discrete_Weibull: $f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!$
Log Gamma: $f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!$
Generalized Gamma: $f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!$
Noncentral-Beta: $f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!$
Inverse Gausian: $f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!$
Noncentral_chi-square: $f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!$
Standard Wald: $f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!$
Inverted Beta: $f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!$
Arctangent: $f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!$
Makeham: $f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta (\kappa^x-1)}{log(\kappa)}). x>0 \!$
Hypoexponential: $f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!$
Doubly Noncentral t: $\!$
Hyperexponential: $f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!$
Muth: $f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!$
Error: $f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!$
Minimax: $f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!$
Noncentral F: $f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!$
IDB: $f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!$
Standard Power: $f(x) = \beta x^{\beta-1}. 0<x<1 \!$
Rayleigh: $f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!$
Standard Triangular: $f(x) = \begin{cases} x+1, -1<x<0 \\ 1 - x, 0 \leq x<1 \end{cases} \!$
Doubly noncentral F: $f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!$
Power: $f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!$
Weibull: $f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!$
Log-logistic: $\!$
TSP: $\!$
Extreme value: $\!$
Lomax: $\!$
von Mises: $\!$
Generalized Pareto: $\!$
Triangular: $\!$
Kolmogorov-Smirnov: $\!$
Exponential Power: $\!$

Transformations

Standard Normal to General Normal Transformation: $\mu+\sigma\times X$
General Normal to Standard Normal Transformation: $X-\mu \over \sigma$
Standard Normal to Chi Transformation: $|\ X |$
Standard Normal to Chi-Square Transformation: $\sum_{k=1}^{\nu} X_k^2$
Gamma to General Normal Transformation: $\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty$
Gamma to Exponential Transformation: The special case of ${\Gamma}(k=1, \theta=1/\lambda)\,$ is equivalent to exponential $E x p (λ)$ .
Gamma to Beta Transformation: $X_1 \over X_1 + X_2$ .
Student's T to Standard Normal Transformation: $n\longrightarrow\infty$
Student's T to Cauchy Transformation: $n=1 \$
Cauchy to General Cauchy Transformation: $a + \alpha\times X$
General Cauchy to Cauchy Transformation: $a=0; \alpha=1 \$
Fisher's F to Student's T: $\sqrt X$
Student's T to Fisher's F: $X 2$
Bernoulli to Binomial Transformation: $\sum X_i$ (iid)
Binomial to Bernoulli Transformation: $\begin{pmatrix} n = 1 \end{pmatrix}$
Binomial to General Normal Transformation: $\begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix}$
Binomial to Poisson Transformation: $\begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix}$
Multinomial to Binomial Transformation: $\begin{vmatrix} k=2 \end{vmatrix}$
Negative Binomial to Geometric Transformation: $\begin{pmatrix} r = 1 \end{pmatrix}$
Erlang to Exponential Transformation: $\begin{pmatrix} k = 1 \end{pmatrix}$
Erlang to Chi-Square Transformation: $\begin{pmatrix} \alpha = 2 \end{pmatrix}$
Laplace to Exponential Transformation: $\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}$
Exponential to Laplace Transformation: $x_1 - x_2 \$
Beta to Arcsine Transformation: $\alpha = \beta = \frac{1}{2}$
Noncentral Student's T to Normal Transformation: $Z=\lim_{\nu\to\infty}T$
Noncentral Student's T to Student's T Transformation: $\mu = 0 \$
Standard Uniform to Pareto Transformation: $\lambda X ^{-1/K} \$
Standard Uniform to Benford Transformation: $10^X \$
Standard Uniform to Exponential Transformation: $n(1-X_{(n)}), n -> \infty$
Standard Uniform to Log Logistic Transformation: $\frac{1}{\lambda}(\frac{1-X}{X})^{1/k}$
Standard Uniform to Standard Triangular Transformation: $X 1 - X 2$
Standard Uniform to Logistic Exponential Transformation: $\frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda}$
Standard Uniform to Beta Transformation: If X has a standard uniform distribution, $Y = 1 - X^{1/n} \$ has a beta distribution
Beta to Standard Uniform Transformation: $β = γ = 1$
Continuous Uniform to Standard Uniform Transformation: $a = 0, b = 1 \$
Pareto to Exponential: $log(X/\lambda) \$
Logistic Exponential to Exponential: $\beta = 1 \$
Zipf to Discrete Uniform: $a = 0, a = 1, b = n \$
Discrete Uniform to Rectangular: $a = 0, b = n \$
Poisson to Normal: $\sigma ^2 = \mu , \mu \to \infty$
Binomial to Poisson: $\mu = np, \mu \to \infty$
Gamma to Inverted Gamma: $\frac{1}{X}$
Fisher-Tippett to Gumbel: $\mu = 0, \beta = 1 \$
Hypergeometric to Binomial: $p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \$
Log-Normal to Normal: $log(X) \$
Normal to Log-Normal: $e^X \$
Log-Normal to Gibrat's: $\mu = 0, x = 1 \$
Cauchy to Standard Cauchy: $\gamma = 1, x_0 = 0 \$
Standard Cauchy to Cauchy: $x_0 + \gamma X \$
Standard Cauchy to Hyperbolic Secant: $\frac{log|x|}{\pi} \$
Beta to Standard Power: $\alpha=\beta, \beta=1 \$

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 * [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0  \!</math>
 * [http://socr.ucla.edu/htmls/dist/Power.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math>
-* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math>  \!</math>
+* [http://socr.ucla.edu/htmls/dist/Weibull.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0  \!</math>
 * [http://socr.ucla.edu/htmls/dist/Log-logistic.html Log-logistic]: <math>  \!</math>
 * [http://socr.ucla.edu/htmls/dist/TSP.html TSP]: <math>  \!</math>

Formulas

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Revision as of 04:48, 27 April 2010

Probability Density Functions (PDFs)

Transformations

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