Multinomial Distribution

The Multinomial distribution generalizes the Binomial: each trial can land in one of $k$ categories (not just success/failure).

Setup

• You run $n$ independent trials.
• Each trial falls into exactly one category $1,2,\dots,k$.
• Category probabilities are $p_1,\dots,p_k$ with $\sum_{i=1}^k p_i = 1$.
• Let $X_i$ be the number of outcomes in category $i$.

Then:

PMF

For counts $x_1,\dots,x_k$ that sum to $n$:

Intuition

• The factorial term counts how many sequences produce the same category totals.
• The product term is the probability of any one specific sequence with those totals.

Relationship to Binomial

If $k=2$ (category A vs not-A), the Multinomial collapses to a Binomial.

Key facts

• Means: $E[X_i] = np_i$
• Variances: $\mathrm{Var}(X_i) = np_i(1-p_i)$
• Covariances: $\mathrm{Cov}(X_i,X_j) = -np_ip_j$ for $i\neq j$

That negative covariance matters: if one count goes up, the others must “make room” because totals sum to $n$.

When to use

Use Multinomial when you count outcomes across multiple categories with replacement / independent trials:

• Dice rolls (1–6), survey responses (A/B/C/D), product segments, etc.

If you sample without replacement from a finite population, you’re in Hypergeometric territory.

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