Distribution cheat sheet

I can never remember the form of distributions so here is a cheat sheet for the density functions of some commonly used ones.

Binomial: $\theta \sim {\rm Bin}(n,p)$

$\displaystyle p(\theta) = \left( \begin{array}{c} n \\ \theta \end{array}\right)p^\theta(1-p)^{n-\theta}$

Multinomial: $\theta \sim {\rm Multin}(n;p_1,\dots,p_k), \theta_j = 0,1,2,\dots n$

$\displaystyle p(\theta) = \left( \begin{array}{c} n \\ \theta_1\theta_2\cdots\theta_k \end{array}\right)p^\theta_1\cdots p^\theta_k, \quad \sum_{j=1}^k\theta_j = n$

Negative binomial: $\theta \sim \mbox{Neg-bin}(\alpha,\beta)$

$\displaystyle p(\theta)= \left( \begin{array}{c} \theta+\alpha-1 \\ \alpha -1 \end{array}\right)\left(\frac{\beta}{\beta+1}\right)^\alpha \left(\frac{1}{\beta+1}\right)^\theta$

Poisson: $\theta \sim {\rm Poisson}(\lambda)$

$\displaystyle p(\theta)=\frac{\lambda^\theta e^{-\lambda}}{\theta!}$

Normal: $\theta \sim {\rm N}(\mu,\sigma^2)$

$\displaystyle p(\theta)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\frac{-(\theta-\mu)^2}{2\sigma^2}}$

Multivariate normal: $\theta \sim {\rm N}(\mu,\Sigma), \theta \in R^d$

$\displaystyle p(\theta)=(2\pi)^{-d/2}|\Sigma|^{-1/2}\exp{\left(-\frac{1}{2}(\theta-\mu)^T\Sigma^{-1}(\theta-\mu)\right)}$

Gamma: $\theta \sim {\rm Gamma}(\alpha,\beta)$

$\displaystyle p(\theta) = \frac{ \beta^\alpha\theta^{\alpha-1} e^{-\beta\theta}}{\Gamma(\alpha)}$

Inverse gamma: $\theta \sim \mbox{Inv-gamma} (\alpha,\beta)$

$\displaystyle p(\theta) = \frac{ \beta^{\alpha} \theta^{-(\alpha +1)} e^{-\beta/\theta}}{\Gamma(\alpha)}$

Beta: $\theta \sim {\rm Beta}(\alpha,\beta)$

$\displaystyle p(\theta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\theta^{\alpha -1}(1-\theta)^{\beta -1}$

Student t: $\theta \sim t_\nu(\mu,\sigma^2)$

$p(\theta)=\frac{\Gamma((\nu+1)/2)}{\Gamma(\nu/2)\sqrt{\nu\pi\sigma}}\left(1+\frac{1}{\nu}\left(\frac{\theta-\mu}{\sigma}\right)^2\right)^{-(\nu+1)/2}$

Cauchy: $\theta \sim t_1(\mu,\sigma^2)$

Chi-square: $\theta \sim \chi_\nu^2 = {\rm Gamma}(\nu/2,1/2)$

One thought on “Distribution cheat sheet”

nkint says:

February 10, 2011 at 18:18

nice summary!

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Carson C. Chow

Distribution cheat sheet

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