A Distribution Tables

A.1 Discrete Distributions

Distribution of \(X\)	\(f(x)\)	Support	\(E[X]\)	\(\text{Var}[X]\)
Hypergeometric\((n, N_1, N_0)\)	\(\displaystyle \frac{\binom{N_1}{x} \binom{N_0}{n-x}}{\binom{N}{n}}\)	\(x=0, 1, \ldots, n\)	\(n \frac{N_1}{N}\)	\(n \frac{N_1}{N} \frac{N_0}{N} \left(1 - \frac{n-1}{N-1}\right)\)
Binomial\((n, N_1, N_0)\)	\(\displaystyle \frac{\binom{n}{x} N_1^x N_0^{n-x}}{N^n}\)	\(x=0, 1, \ldots, n\)	\(n \frac{N_1}{N}\)	\(n \frac{N_1}{N} \frac{N_0}{N}\)
Binomial\((n, p)\)	\(\binom{n}{x} p^x (1-p)^{n-x}\)	\(x=0, 1, \ldots, n\)	\(np\)	\(np(1-p)\)
Geometric\((p)\)	\((1-p)^{x-1} p\)	\(x=1, 2, \ldots\)	\(\frac{1}{p}\)	\(\frac{1-p}{p^2}\)
NegativeBinomial\((r, p)\)	\(\binom{x-1}{r-1} (1-p)^{x-r} p^r\)	\(x=r, r+1, \ldots\)	\(\frac{r}{p}\)	\(\frac{r(1-p)}{p^2}\)
Poisson\((\mu)\)	\(e^{-\mu} \frac{\mu^x}{x!}\)	\(x=0, 1, 2, \ldots\)	\(\mu\)	\(\mu\)

A.2 Continuous Distributions

Distribution of \(X\)	\(f(x)\)	Support	\(E[X]\)	\(\text{Var}[X]\)
Uniform\((a, b)\)	\(\frac{1}{b-a}\)	\(a < x < b\)	\(\frac{a+b}{2}\)	\(\frac{(b-a)^2}{12}\)
Exponential\((\lambda)\)	\(\lambda e^{-\lambda x}\)	\(0 < x < \infty\)	\(\frac{1}{\lambda}\)	\(\frac{1}{\lambda^2}\)
Gamma\((r, \lambda)\)	\(\frac{\lambda^r}{(r-1)!}x^{r-1} e^{-\lambda x}\)	\(0 < x < \infty\)	\(\frac{r}{\lambda}\)	\(\frac{r}{\lambda^2}\)
Normal\((\mu, \sigma)\)	\(\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}\)	\(-\infty < x < \infty\)	\(\mu\)	\(\sigma^2\)