Multinomial distribution

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, the multinomial distribution is a generalization of the <a href="/facts/Binomial_distribution/UMoFMjDj">binomial distribution</a>. For example, it models the probability of counts for each side of a k-sided die rolled n times. For n <a href="/facts/Statistical_independence/NUzQtnUL">independent</a> trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.
When k is 2 and n is 1, the multinomial distribution is the <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli distribution</a>. When k is 2 and n is bigger than 1, it is the <a href="/facts/Binomial_distribution/UMoFMjDj">binomial distribution</a>. When k is bigger than 2 and n is 1, it is the <a href="/facts/Categorical_distribution/xpc6RWM2">categorical distribution</a>. The term "multinoulli" is sometimes used for the categorical distribution to emphasize this four-way relationship (so n determines the suffix, and k the prefix).
The <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli distribution</a> models the outcome of a single <a href="/facts/Bernoulli_trial/7fm8snCf">Bernoulli trial</a>. In other words, it models whether flipping a (possibly <a href="/facts/Fair_coin/vTBk29k7">biased</a>) coin one time will result in either a success (obtaining a head) or failure (obtaining a tail). The <a href="/facts/Binomial_distribution/UMoFMjDj">binomial distribution</a> generalizes this to the number of heads from performing n independent flips (Bernoulli trials) of the same coin. The multinomial distribution models the outcome of n experiments, where the outcome of each trial has a <a href="/facts/Categorical_distribution/xpc6RWM2">categorical distribution</a>, such as rolling a (possibly <a href="/facts/Loaded_dice/qBnylzwJ">biased</a>) k-sided die n times.
Let k be a fixed finite number. Mathematically, we have k possible mutually exclusive outcomes, with corresponding probabilities p1, ..., pk, and n independent trials. Since the k outcomes are mutually exclusive and one must occur we have pi ≥ 0 for i = 1, ..., k and 
 
 
 
 
 ∑
 
 i
 =
 1
 
 
 k
 
 
 
 p
 
 i
 
 
 =
 1
 
 
 {\displaystyle \sum _{i=1}^{k}p_{i}=1}
 
. Then if the random variables Xi indicate the number of times outcome number i is observed over the n trials, the vector X = (X1, ..., Xk) follows a multinomial distribution with parameters n and p, where p = (p1, ..., pk). While the trials are independent, their outcomes Xi are dependent because they must sum to n.

Multinomial distribution open-in-new

Multinomial distribution