Hypergeometric Distribution

Definition

$X\sim \mathrm{Hypergeom}(N,K,n)$ where $N$ is the population size, $K$ is the number of success, and $n$ is the number of trials,

Distribution that describes the probability of $k$ successes in $n$ draws, without replacement, from a population $N$ that contains $K$ successes.

Draws without replacement version of Binomial Distribution

Properties

PDF

$f(x)=\frac{\left(\genfrac{}{}{0ex}{}{K}{k}\right)\left(\genfrac{}{}{0ex}{}{N-K}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)},x\in \{0,1,\dots ,n\}$

Mean

$E(X)=n\frac{K}{N}$

Variance

$\mathrm{Var}(X)=n\frac{K}{N}\frac{N-K}{N}\frac{N-n}{N-1}$