Wednesday, 27 November 2013

Probability Distributions Tables

Below are some tables detailing the Density Equation, parameters, support and Cumulative Distribution Function (CDF) for a few commonly used probability distributions.

In the tables that follow, the error function $erf(x)$ is \begin{equation} erf(x) = \frac{2}{\sqrt{\pi}}\int_0^xexp(-t^2)dt \nonumber \end{equation} the Gamma function $\Gamma(a)$ is the following integral \begin{equation} \Gamma(a)=\int_0^\infty t^{(a-1)}e^{-t}dt \nonumber \end{equation} the lower incomplete Gamma function $\gamma(a,x)$ is \begin{equation} \gamma(a,x)=\int_0^x t^{(a-1)}e^{-t}dt \nonumber \end{equation} The normalised (or regularized) incomplete Beta function $I_x(a,b)$ is \begin{equation} I_x(a,b)=\frac{B_x(a,b)}{B(a,b)} \nonumber \end{equation} where $x\leq 1$, and $B(a,b)$ and $B_x(a,b)$ are (respectively) \begin{equation} B(a,b)=\int_0^1t^{(a-1)}(1-t)^{(b-1)}dt \nonumber \end{equation} and \begin{equation} B_x(a,b)=\int_0^xt^{(a-1)}(1-t)^{(b-1)}dt \nonumber \end{equation}

Distribution Beta
Density Equation $\Large \frac{x^{(\alpha - 1)}(1-x)^{\beta-1}}{B(\alpha,\beta)}$
Parameters $\alpha>0,\beta>0$
Support $0\leq x \leq 1$
CDF$I_x(\alpha,\beta)$

Distribution Binomial
Density Equation ${n \choose k} p^k(1-p)^{n-k}$
Parameters $n \in \{0,1,2,..\},0< p < 1$
Support $k \in\{0,1,2,..\}$
CDF$I_{1-p}(n-k,1+k)$

Distribution Cauchy-Lorenz
Density Equation $\Large \frac{1}{\pi\gamma[1+{(\frac{x-x_0}{\gamma})}^2]}$
Parameters $-\infty < x_0 < +\infty,\gamma > 0$
Support $-\infty < x < +\infty$
CDF$\frac{1}{\pi}atan(\frac{x-x_0}{\gamma})+\frac{1}{2}$

Distribution Chi-squared
Density Equation $\Large \frac{1}{2^{k/2}\Gamma(k/2)}x^{\frac{k}{2}-1}e^{-x/2}$
Parameters $k \in \{0,1,2,..\}$
Support $0\leq x < \infty$
CDF$\Large \frac{1}{\Gamma(k/2)}\gamma(\frac{k}{2},\frac{x}{2})$

Distribution F-distribution
Density Equation $\Large \frac{\sqrt{\frac{(d_1x)^{d_1}d_2^{d_2}}{(d_1x+d_2)^{d_1+d_2}}}}{xB(d_1/2,d_2/2)}$
Parameters $d_1>0,d_2>0$
Support $0\leq x<\infty$
CDF$\Large I_{\frac{d_1x}{d_1x+d_2}}(\frac{d_1}{2},\frac{d_2}{2})$

Distribution Gamma
Density Equation $\Large \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}$
Parameters $\alpha>0,\beta>0$
Support $0\leq x<\infty$
CDF$\Large \frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)$

Distribution Gaussian
Density Equation $\Large \frac{1}{\sigma\sqrt{2\pi}}exp(-\frac{(x-\mu)^2}{2\sigma^2})$
Parameters $-\infty<\mu<\infty,\sigma>0$
Support $-\infty < x < \infty$
CDF$\Large \frac{1}{2}(1+erf(\frac{x-\mu}{\sigma\sqrt{2}}))$

Distribution Hypergeometric
Density Equation $\Large \frac{{r \choose k}{N-r \choose n-k}}{{N \choose n}}$
Parameters $0\leq k \leq r$, $0\leq (n-k) \leq (N-r)$
Support $k\in\{0,1,2,..\}$
CDFSee below

The CDF of the Hypergeometric distribution is given by the following equation:- \begin{equation} CDF = 1 - \frac{{n \choose k+1}{N-n \choose r-k-1}}{{N \choose r}}(1+\sum_{m=1}^{\infty}\frac{(k+m-r)(k+m-n)}{(k+m+1)(N+k+1+m-r-n)}\frac{1}{(m - 1)!}) \nonumber \end{equation}
Distribution Laplace
Density Equation $\Large \frac{1}{2b}exp(-\frac{|x-\mu|}{b})$
Parameters $-\infty < \mu < \infty,b > 0$
Support $-\infty < x < \infty$
CDF$\Large \frac{1}{2}exp(\frac{x-\mu}{b})$ if $x<\mu$
$\Large 1-\frac{1}{2}exp(\frac{\mu-x}{b})$ if $x\geq \mu$

Distribution Logistic
Density Equation $\Large \frac{e^{-\frac{(x-\mu)}{s}}}{s(1+e^{-\frac{x-\mu}{s}})^2}$
Parameters $-\infty < \mu < \infty,s>0$
Support $-\infty < x < \infty$
CDF$\Large \frac{1}{(1+e^{\frac{x-\mu}{s}})^2}$

Distribution Log-Normal
Density Equation $\Large \frac{1}{x\sqrt{2\pi\sigma^2}}exp(-\frac{(ln(x)-\mu)^2}{2\sigma^2})$
Parameters $\sigma > 0,-\infty < \mu < \infty$
Support $0 < x < \infty$
CDF$\Large \frac{1}{2}(1+erf(\frac{ln(x)-\mu}{\sigma\sqrt{2}}))$

Distribution Pascal
Density Equation ${k+r-1 \choose k}(1-p)^rp^k$
Parameters $0 < p < 1,r > 0$
Support $k\in\{0,1,2,..\}$
CDF$1-I_p(k+1,r)$

Distribution Poisson
Density Equation $\Large \frac{\lambda^k}{k!}e^{-\lambda}$
Parameters $\lambda > 0$
Support $k\in\{0,1,2,...\}$
CDF$\Large \frac{\Gamma(k+1,\lambda)}{k!}$

Distribution Rayleigh
Density Equation $\Large \frac{x}{\sigma^2}exp(-\frac{x^2}{2\sigma^2})$
Parameters $\sigma > 0$
Support $0\leq x < \infty$
CDF$\Large 1-exp(-x^2/2\sigma^2)$

Distribution Student-t
Density Equation$\Large \frac{\Gamma(\frac{v+1}{2})}{\sqrt{v\pi}\Gamma(\frac{v}{2})}(1+\frac{x^2}{v})^{\frac{v+1}{2}}$
Parameters $v\in{1,2,3,..}$
Support $-\infty < x < \infty$
CDF$\Large 1-\frac{1}{2}I_{x(t)}(\frac{v}{2},\frac{1}{2})$ for $t>0$ where $\Large x(t)=\frac{v}{t^2+v}$

Distribution Weibull
Density Equation$\Large \frac{k}{\lambda}(\frac{x}{\lambda})^{(k-1)}exp(-(\frac{x}{\lambda})^k)$
Parameters $\lambda > 0,k > 0$
Support $0\leq x < \infty$
CDF$\Large 1-exp\{-(\frac{x}{\lambda})^k\}$