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)


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

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


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