Multinomial Distribution

1 min read Last updated Sat Jun 27 2026 08:46:52 GMT+0000 (Coordinated Universal Time)

Suppose the random variables $X_1, X_2, \ldots, X_k$ represent the number of times each outcome $\theta_1, \theta_2, \ldots, \theta_k$ occurs in $n$ independent trials.

These random variables are said to follow a multinomial distribution with parameters $n$ and $\mathbf{p} = (p_1, p_2, \ldots, p_k)$ iff the following conditions are met:

\sum_{i=1}^{k} p_i = 1

\sum_{i=1}^{k} x_i = n

P(X_1 = x_1, X_2 = x_2, \ldots, X_k = x_k) = \frac{n!}{x_1! \, x_2! \, \cdots \, x_k!} \, p_1^{x_1} \, p_2^{x_2} \, \cdots \, p_k^{x_k}

\binom{n}{x_1, x_2, \ldots, x_k}= \frac{n!}{x_1! \, x_2! \, \cdots \, x_k!}

Mean and Variance

For each outcome $i$ :

\mu_{X_i} = np_i

\sigma^2_{X_i} = np_i(1 - p_i)

Relation to Binomial

The binomial distribution is a special case with $k = 2$ outcomes (success and failure).

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