Hypergeometric Distribution

Compact study note.

Summary

The hypergeometric distribution counts successes in fixed-size samples drawn without replacement from finite populations.^[1]

$X \sim Hypergeometric (N, K, n)$ when one population of $N$ items has $K$ successes and $n$ items are sampled without replacement.

Population size and number of successes are fixed. Sampling is without replacement.

$N \in {1, 2, \dots}$ , $0 \leq K \leq N$ , and $0 \leq n \leq N$ .

$max (0, n - (N - K)) \leq k \leq min (n, K)$ .

$P (X = k) = \frac{(\binom{K}{k}) (\binom{N - K}{n - k})}{(\binom{N}{n})}$ on the valid support.

Usually evaluated by summing the PMF over valid integers.

$E [X] = n K / N$ and $Var (X) = n (K / N) (1 - K / N) (N - n) / (N - 1)$ for $N > 1$ .

Hypergeometric is the without-replacement analogue of binomial sampling.

From 50 items with 20 successes, sample 10. Then $P (X = 4) = (\binom{20}{4}) (\binom{30}{6}) / (\binom{50}{10})$ .

Using a binomial model when draws are without replacement and the population is not large.
Allowing impossible $k$ values outside the support.

OpenStax, Introductory Statistics 2e, "Chapter 4: Discrete Random Variables", https://openstax.org/books/introductory-statistics-2e/pages/4-introduction ↩︎