Hypergeometric

Hypergeometric Random Variables

Definition:[Hypergeometric Random Variable]
Let $N,D$ and $n$ be positive integers with $n \leq \min (D,N-D)$ , then a random variable with pmf,

$\begin{align*} p(x) &= P(X = x) = \begin{cases} \frac{\dbinom{D}{x} \dbinom{N - D}{n - x}}{\dbinom{N}{n}} & \text{if } x \in \set{0,1,...,n} \\ 0 & \text{otherwise} \end{cases} \end{align*}$

is called a hypergeometric random variable. It has the following:

$\begin{align*} E[X] &= \frac{nD}{N} \\ V[X] &= \frac{nD(N - D)}{N^2}\left( 1 - \frac{n -1 }{N - 1} \right) \end{align*}$