Table Summary of Probability Distributions and Properties

Name	Type	Range	Density Function	Mean	Variance
Binomial Distribution	Discrete	$x \in [0, n]$	$f(x)=\dbinom{n}{x} p^x (1−p)^{n−x}$	$np$	$np(1-p)$
Bernoulli Distribution	Discrete	$x = 0, 1$	$f(x) =,\left\{,\begin{array}{lll},p & \qquad k = 1 \\,1 − p & \qquad k = 0 \\,\end{array},\right.$	$p$	$p(1-p)$
Uniform Distribution	Discrete	$x \in [a,b]$	$f(x) = \frac{1}{b − a + 1}$	$\frac{a+b}{2}$	$\frac{[(b − a + 1)^2 − 1]}{12}$
Uniform Distribution	Continuous	$x \in (a,b)$	$f(x) = \frac{1}{b − a}$	$\frac{a+b}{2}$	$\frac{[(b - a)^2]}{12}$
Poisson Distribution	Discrete	$x \in [0, \infty)$	$f(x)=\dfrac{e^{−\lambda} \lambda^x}{x!}$	$\lambda$	$\lambda$
Geometric Distribution	Discrete	$x \in (0, \infty)$	$f(x) = p(1−p)^{x−1}$	$\frac{1}{p}$	$\frac{1-p}{p^2}$
Negative Binomial Distribution	Discrete	$x \in [0,r]$	$f(x) = \binom{n − 1}{r − 1} p^r (1 − p)^{n − r}$	$\frac{r(1-p)}{p}$	$\frac{r(1-p)}{p^2}$
Hypergeometric Distribution	Discrete	$x \leq G$ $n - x \leq N-m$	$f(x)=\dfrac{\dbinom{m}{x}\dbinom{N−m}{n−x}}{\dbinom{N}{n}}$	$\frac{nG}{N}$	$\frac{nG}{N} \frac{N-G}{N} \frac{N-n}{N-1}$
Beta Distribution	Continuous	$x \in [0,1]$	$f(x) = \frac{\Gamma(r + s)}{\Gamma(r,)\Gamma(s)} x^{r−1} (1 − x)^{s−1}$	$\frac{r}{r+s}$	$\frac{rs}{(r+s)(r+s+1)}$
Exponential Distribution	Continuous	$x \ in [0, \infty)$	$f(x)=\lambda e^{−x \lambda}$	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$
Gamma Distribution	Continuous	$x \in (0, \infty)$	$f(x)=\dfrac{1}{(r−1)! \lambda^r} e^{−x/\lambda} x^{r−1}$	$\frac{r}{\lambda}$	$\frac{r}{\lambda}$
Normal (Gaussian) Distribution	Continuous	$x \in (-\infty, \infty)$	$f(x)=\dfrac{1}{\sigma \sqrt{2\pi}} \text{exp}\left\{−\dfrac{1}{2} \left(\dfrac{x−\mu}{\sigma}\right)^2\right\}$	$\mu$	$\sigma^2$
Chi-Square Distribution	Continuous	$x \ in [0, \infty)$	$f(x)=\dfrac{1}{\Gamma (n/2) 2^{n/2}}x^{n/2−1}e^{−x/2}$	$n$	$2n$