A multivariate, discrete probability distribution used to describe the results of a random experiment where each of $n$ outcomes are placed into one of $k$ nominal categories.

Overview

The multinomial distribution is a discrete probability distribution used to describe the results of a random experiment where each of $n$ outcomes are placed into one of $k$ nominal categories. It can be thought of as the generalization of the binomial distribution. The binomial distribution is a special case of the multinomial distribution where there are only $k=2$ categories.

The probability mass function (pmf) of the distribution is parametrized by $p_i$, the probabilities of $x_i$ ($i=1,2,\ldots k$) outcomes being placed in the $i^\text{th}$ category. The pmf $P(x_i,n;p_i)$ has the following form:

$$ \left\{ \begin{array}{l l} \frac{n!}{x_1!x_2!\ldots x_n!} p_1^{x_1} p_2^{x_2} \ldots p_n^{x_n} & \quad \text{if $\sum_{n=1}^k x_i =n$ }\\ ~ \\ ~ \\ 0 & \quad \text{otherwise} \end{array} \right. $$

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