Hypergeometric Distribution Reference Cheat Sheet

Key Facts

• The hypergeometric distribution applies when sampling $n$ items without replacement from a finite population of size $N$ containing $K$ successes.
• If $X$ counts successes in the sample, write $X \sim \operatorname{Hypergeometric}(N,K,n)$.
• The probability mass function is $P(X=x)=\frac{\binom{K}{x}\binom{N-K}{n-x}}{\binom{N}{n}}$.
• The support is $\max(0,n-(N-K)) \le x \le \min(n,K)$, where $x$ must be an integer.
• The mean is $E(X)=n\frac{K}{N}$.
• The variance is $\operatorname{Var}(X)=n\frac{K}{N}\left(1-\frac{K}{N}\right)\frac{N-n}{N-1}$.
• The factor $\frac{N-n}{N-1}$ is the finite population correction that makes the variance smaller than the binomial variance.
• For large $N$ with small sampling fraction $\frac{n}{N}$, a binomial approximation with $p=\frac{K}{N}$ is often reasonable.

Vocabulary

Population size

The total number of objects in the finite population, usually denoted by $N$.

Success states

The number of population objects that have the characteristic being counted, usually denoted by $K$.

Sample size

The number of objects drawn from the population without replacement, usually denoted by $n$.

Support

The set of possible integer values of $X$, from $\max(0,n-(N-K))$ to $\min(n,K)$.

Probability mass function

A formula that gives $P(X=x)$ for each possible value of a discrete random variable.

Finite population correction

The multiplier $\frac{N-n}{N-1}$ that adjusts variance because draws are made without replacement.