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The multinomial is an example of a parametric distribution for multiple random variables.

Say you perform $n$ independent trials of an experiment where each trial results in one of $m$ outcomes, with respective probabilities: $p_1, p_2, \dots , p_m$ (constrained so that $\sum_i p_i = 1$). Define $X_i$ to be the number of trials with outcome $i$. A multinomial distribution is a closed form function that answers the question: What is the probability that there are $c_i$ trials with outcome $i$. Mathematically: \begin{align*} P(X_1=c_1,X_2 = c_2, \dots , X_m = c_m) = { {n} \choose {c_1,c_2,\dots , c_m} }\cdot p_1^{c_1} \cdot p_2^{c_2}\dots p_m^{c_m} \end{align*}

Often people will use the product notation to write the exact same equation: \begin{align*} P(X_1=c_1,X_2 = c_2, \dots , X_m = c_m) = { {n} \choose {c_1,c_2,\dots , c_m} }\cdot \prod_i p_i^{c_i} \end{align*}

Example: A 6-sided die is rolled 7 times. What is the probability that you roll: 1 one, 1 two, 0 threes, 2 fours, 0 fives, 3 sixes (disregarding order). \begin{align*} P(X_1=1,X_2 = 1&, X_3 = 0,X_4 = 2,X_5 = 0,X_6 = 3) \\&= \frac{7!}{2!3!}\left(\frac{1}{6}\right)^1\left(\frac{1}{6}\right)^1\left(\frac{1}{6}\right)^0\left(\frac{1}{6}\right)^2\left(\frac{1}{6}\right)^0\left(\frac{1}{6}\right)^3\\ &=420\left(\frac{1}{6}\right)^7 \end{align*}

The multinomial is especially popular because of its use as a model of language. For a full example see the Federalist Paper Authorship example.