Fast Distributions

This file exists as a cheat sheet to get information.

Distribution	PDF/CDF	Expectation $E[X]$	Variance $\text{Var}(X)$	Second Moment $E[X^2]$
Bernoulli$(p)$	PDF: $p^x(1-p{)}^{1-x},x\in \{0,1\}$	$p$	$p(1-p)$	$p$
Binomial$(n,p)$	PDF: $\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$	$np$	$np(1-p)$	$np(1-p)+(np{)}^2$
Geometric$(p)$	PDF: $(1-p{)}^{k-1}p,k\in \{1,2,\dots \}$	$1/p$	$(1-p)/p^2$	$(2-p)/p^2$
Poisson$(\lambda )$	PDF: $\frac{{\lambda }^k{e}^{-\lambda }}{k!},k\in \{0,1,2,\dots \}$	$\lambda$	$\lambda$	$\lambda +{\lambda }^2$
Uniform$(a,b)$	PDF: $\frac{1}{b-a},a\le x\le b$	$\frac{a+b}{2}$	$\frac{(b-a{)}^2}{12}$	$\frac{a^2+ab+b^2}{3}$
Exponential$(\lambda )$	PDF: $\lambda e^{-\lambda x},x>0$	$1/\lambda$	$1/{\lambda }^2$	$2/{\lambda }^2$
Normal$(\mu ,{\sigma }^2)$	PDF: $\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$	$\mu$	${\sigma }^2$	${\mu }^2+{\sigma }^2$
Gamma$(k,\lambda )$	PDF: $\frac{{\lambda }^k{x}^{k-1}{e}^{-\lambda x}}{\Gamma (k)},x>0$	$k/\lambda$	$k/{\lambda }^2$	$(k+1)/{\lambda }^2$