In probability theory, the **multinomial distribution** is a generalization of the binomial distribution.

The binomial distribution is the probability distribution of the number of "successes" in *n* independent Bernoulli trials, with the same probability of "success" on each trial. In a multinomial distribution, the analog of the Bernoulli distribution is the categorical distribution, where each trial results in exactly one of some fixed finite number *k* of possible outcomes, with probabilities *p*_{1}, ..., *p*_{k} (so that *p*_{i} ≥ 0 for *i* = 1, ..., *k* and ), and there are *n* independent trials. Then let the random variables *X*_{i} indicate the number of times outcome number *i* was observed over the *n* trials. The vector *X* = (*X*_{1}, ..., *X*_{k}) follows a multinomial distribution with parameters *n* and **p**, where **p** = (*p*_{1}, ..., *p*_{k}).

Note that, in some fields, such as natural language processing, the categorical and multinomial distributions are conflated, and it is common to speak of a "multinomial distribution" when a categorical distribution is actually meant. This stems from the fact that it is sometimes convenient to express the outcome of a categorical distribution as a "1-of-K" vector (a vector with one element containing a 1 and all other elements containing a 0) rather than as an integer in the range ; in this form, a categorical distribution is equivalent to a multinomial distribution over a single observation.

Read more about Multinomial Distribution: Properties, Example, Sampling From A Multinomial Distribution, To Simulate A Multinomial Distribution, Related Distributions

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“Classical and romantic: private language of a family quarrel, a dead dispute over the *distribution* of emphasis between man and nature.”

—Cyril Connolly (1903–1974)