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$$(\mu, \sigma)$$ $$\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$$ $$-\infty < x < \infty$$ $$\mu$$ $$\sigma^2$$