$\DeclareMathOperator{\p}{P}$ $\DeclareMathOperator{\P}{P}$ $\DeclareMathOperator{\c}{^C}$ $\DeclareMathOperator{\or}{ or}$ $\DeclareMathOperator{\and}{ and}$ $\DeclareMathOperator{\var}{Var}$ $\DeclareMathOperator{\Var}{Var}$ $\DeclareMathOperator{\Std}{Std}$ $\DeclareMathOperator{\E}{E}$ $\DeclareMathOperator{\std}{Std}$ $\DeclareMathOperator{\Ber}{Bern}$ $\DeclareMathOperator{\Bin}{Bin}$ $\DeclareMathOperator{\Poi}{Poi}$ $\DeclareMathOperator{\Uni}{Uni}$ $\DeclareMathOperator{\Geo}{Geo}$ $\DeclareMathOperator{\NegBin}{NegBin}$ $\DeclareMathOperator{\Beta}{Beta}$ $\DeclareMathOperator{\Exp}{Exp}$ $\DeclareMathOperator{\N}{N}$ $\DeclareMathOperator{\R}{\mathbb{R}}$ $\DeclareMathOperator*{\argmax}{arg\,max}$ $\newcommand{\d}{\, d}$

Multinomial


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.