Multinomial Distribution¶

Definition¶

Let $n\ge0$ be an integer and $\boldsymbol{p}=(p_1,\dots,p_k)$ with $p_j\ge0$ and $\displaystyle \sum_{j=1}^kp_j=1$ $1$ . A random vector $\boldsymbol{X}=(X_1,\dots,X_k)$ has a multinomial distribution, written $\boldsymbol{X}\sim\operatorname{Mult}(n,\boldsymbol{p})$ $Mult (n, p)$ , if for all integers $n_j\ge0$ with $\displaystyle \sum_jn_j=n$

\begin{aligned} \mathbb{P}(\boldsymbol{X}=\boldsymbol{n})=\frac{n!}{n_1!\cdots n_k!}p_1^{n_1}\cdots p_k^{n_k}, \end{aligned}

with $0^0=1$ when a category probability is zero; the PMF is zero otherwise. The margins are binomial: $X_j\sim\operatorname{Bin}(n,p_j)$ $Bin (n, p_{j})$ .

The displayed formula is the Joint Probability Mass Function of all category counts. Each single count has a Binomial Distribution, since each trial either lands in that category or does not.

Interpretation¶

A multinomial vector counts outcomes in several categories across a fixed number of independent trials. Its category counts must add to the fixed total.

Ten discs arranged in a triangle, one for each count vector of three trials over three categories, with disc areas proportional to the multinomial probabilities; the vector (1, 1, 1) is highlighted.

The diagram uses $n=3$ trials with category probabilities $(1/2,1/4,1/4)$ . Each disc represents a count vector $(n_1,n_2,n_3)$ whose entries sum to $3$ , and its area is proportional to the probability given by the PMF above. The red disc marks $(1,1,1)$ , with probability $3/16$ , calculated in the first example below. The counts are dependent because their total is fixed.

Key Results¶

$X_j\sim\operatorname{Bin}(n,p_j)$ $Bin (n, p_{j})$ , so $\mathbb{E}X_j=np_j$ and $\operatorname{Var}(X_j)=np_j(1-p_j)$ $p_{j})$ .
For $i\ne j$ $j$ , $\operatorname{Cov}(X_i,X_j)=-np_ip_j$ $- n p_{i} p_{j}$ .

Examples¶

For $n=3$ and category probabilities $(1/2,1/4,1/4)$ , $\mathbb{P}(\boldsymbol{X}=(1,1,1))=6(1/2)(1/4)(1/4)=3/16$ $3/16$ .
For $n=10$ and probabilities $(0.2,0.3,0.5)$ , $\mathbb{E}X_1=2$ and $\operatorname{Cov}(X_1,X_2)=-0.6$ $- 0.6$ .

Prerequisites¶

Sources¶

Nicky van Foreest · Probability Distributions · Week 2

Symbols: sum

Last updated: 14 October 2026 08:21