Hypergeometric Distribution

The hypergeometric distribution models the probability of obtaining a specified number of successes in a fixed number of draws without replacement from a finite population containing a known number of successes and failures.

A hypergeometric random variable is denoted:

\[ X \sim \text{Hypergeometric}(N, M, n) \]

where

• \(N\) = total number of objects available

• \(M\) = total number of possible successes

• \(n\) = sample size

• \(x\) = number of successes in the sample

Probability Mass Function (PMF)

The pmf of the hypergeometric distribution is as follows:

\[\begin{split} p_X(x) = \begin{cases} \frac{\binom{M}{x} \binom{N-M}{n-x}}{\binom{N}{n}} & x=0,1,...,n \\ 0, & \text{otherwise} \end{cases} \end{split}\]

Expectation

The expectation of the hypergeometric distribution is derived as follows.

\[\begin{split} \begin{align} EX &= \sum_{x=0}^n x \ p_X(x) \\ &= \sum_{x=0}^n x\frac{\binom{M}{x} \binom{N-M}{n-x}}{\binom{N}{n}} \\ &= \sum_{x=1}^n x \frac{\frac{M}{x} \binom{M-1}{x-1} \binom{N-M}{n-x}}{\frac{N}{n} \binom{N-1}{n-1}} \\ &= \frac{nM}{N} \sum_{x=0}^n \frac{\binom{M-1}{x-1} \binom{N-M}{n-x}}{\binom{N-1}{n-1}} \\ &= \frac{nM}{N} \sum_{y=0}^{n-1} \frac{\binom{M-1}{y} \binom{N-1-(M-1)}{n-1-y}}{\binom{N-1}{n-1}} \\ &= \frac{nM}{N} \sum_{y=0}^{n-1} p_Y(y) \\ &= \frac{nM}{N} \end{align} \end{split}\]

Note the following:

• In line (3), we the following: \(\binom{M}{x} = \frac{M}{x} \binom{M-1}{x-1}\).

• In line (5), the variable \(x\) is substituted for \(y=x-1\)

• When going from line (5) to line (6), we recognize that the summation is equal to \(1\) since \(Y \sim \text{Hypergeometric}(N-1,M-1,n-1)\)

Variance

Here is the variance of the hypergeometric distribution:

\[ Var(X) = n \frac{M}{N}(1 - \frac{M}{N})(\frac{N-n}{N-1}) \]