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## 13.4 Hypergeometric distribution

### 13.4.1 Probability mass function

If $$a \in \mathbb{N}$$, $$b \in \mathbb{N}$$, and $$N \in \{0,\ldots,a+b\}$$, then for $$n \in \{\max(0,N-b),\ldots,\min(a,N)\}$$, $\text{Hypergeometric}(n~|~N,a,b) = \frac{\normalsize{\binom{a}{n} \binom{b}{N - n}}} {\normalsize{\binom{a + b}{N}}}.$

### 13.4.2 Sampling statement

n ~ hypergeometric(N, a, b)

Increment target log probability density with hypergeometric_lupmf(n | N, a, b).

### 13.4.3 Stan functions

real hypergeometric_lpmf(int n | int N, int a, int b)
The log hypergeometric probability mass of n successes in N trials given total success count of a and total failure count of b

real hypergeometric_lupmf(int n | int N, int a, int b)
The log hypergeometric probability mass of n successes in N trials given total success count of a and total failure count of b dropping constant additive terms

int hypergeometric_rng(int N, int a, int b)
Generate a hypergeometric variate with N trials, total success count of a, and total failure count of b; may only be used in transformed data and generated quantities blocks