Summary of Some Common Distributions

Name and parameters	Possible values	pmf/pdf	cdf	Mean	SD
Discrete
Poisson( $μ$ )	$x = 0, 1, 2, \dots$	$e^{- μ} \frac{μ^{x}}{x!}$	No closed form	$μ$	$\sqrt{μ}$
Binomial( $n$ , $p$ )	$x = 0, 1, 2, \dots, n$	$(\binom{n}{x}) p^{x} (1 - p)^{n - x}$	No closed form	$n p$	$\sqrt{n p (1 - p)}$
Geometric( $p$ )	$x = 1, 2, 3, \dots$	$p (1 - p)^{x - 1}$	$1 - (1 - p)^{x}$	$\frac{1}{p}$	$\sqrt{\frac{1 - p}{p^{2}}}$
Negative Binomial( $r$ , $p$ )	$x = r, r + 1, r + 2, \dots$	$(\binom{x - 1}{x - r}) p^{r} (1 - p)^{x - r}$	No closed form	$\frac{r}{p}$	$\sqrt{\frac{r (1 - p)}{p^{2}}}$
Hypergeometric( $n$ , $N_{1}$ , $N_{0}$ ); $N = N_{1} + N_{0}, p = \frac{N_{1}}{N}$	$x = 0, 1, \dots, n$ (usually)	$\frac{(\binom{N_{1}}{x}) (\binom{N_{0}}{n - x})}{(\binom{N_{1} + N_{0}}{n})}$	No closed form	$n p$	$\sqrt{n p (1 - p) (\frac{N - n}{N - 1})}$
Continuous
Uniform( $a$ , $b$ )	$a < x < b$	$\frac{1}{b - a}$	$\frac{x - a}{b - a}$	$\frac{a + b}{2}$	$\frac{b - a}{\sqrt{12}}$
Exponential( $λ$ )	$x > 0$	$λ e^{- λ x}$	$1 - e^{- λ x}$	$\frac{1}{λ}$	$\frac{1}{λ}$
Normal( $μ$ , $σ$ )	$- \infty < x < \infty$	$\frac{1}{σ \sqrt{2 π}} \exp (- \frac{1}{2} {(\frac{x - μ}{σ})}^{2})$	No closed form	$μ$	$σ$