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## 14.1 Multinomial Distribution

### 14.1.1 Probability Mass Function

If $$K \in \mathbb{N}$$, $$N \in \mathbb{N}$$, and $$\theta \in \text{K-simplex}$$, then for $$y \in \mathbb{N}^K$$ such that $$\sum_{k=1}^K y_k = N$$, $\text{Multinomial}(y|\theta) = \binom{N}{y_1,\ldots,y_K} \prod_{k=1}^K \theta_k^{y_k},$ where the multinomial coefficient is defined by $\binom{N}{y_1,\ldots,y_k} = \frac{N!}{\prod_{k=1}^K y_k!}.$

### 14.1.2 Sampling Statement

y ~ multinomial(theta)

Increment target log probability density with multinomial_lpmf( y | theta) dropping constant additive terms.

### 14.1.3 Stan Functions

real multinomial_lpmf(int[] y | vector theta)
The log multinomial probability mass function with outcome array y of size $$K$$ given the $$K$$-simplex distribution parameter theta and (implicit) total count N = sum(y)

int[] multinomial_rng(vector theta, int N)
Generate a multinomial variate with simplex distribution parameter theta and total count N; may only be used in generated quantities block