# multinomial distribution

Let $\textbf{X}=(X_{1},\ldots,X_{n})$ be a random vector such that

1. 1.

$X_{i}\geq 0$ and $X_{i}\in\mathbb{Z}$

2. 2.

$X_{1}+\cdots+X_{n}=N$, where $N$ is a fixed integer

Then X has a multinomial distribution if there exists a parameter vector $\boldsymbol{\pi}=(\pi_{1},\ldots,\pi_{n})$ such that

1. 1.

$\pi_{i}\geq 0$ and $\pi_{i}\in\mathbb{R}$

2. 2.

$\pi_{1}+\cdots+\pi_{n}=1$

3. 3.

X has a discrete probability distribution function $f_{\textbf{X}}(\boldsymbol{x})$ in the form:

 $f_{\textbf{X}}(\boldsymbol{x})=\frac{N!}{x_{1}!\cdots x_{n}!}\prod_{i=1}^{n}% \pi_{i}^{x_{i}}$

Remarks

• $\operatorname{E}[\textbf{X}]=N\boldsymbol{\pi}$

• $\operatorname{Var}[\textbf{X}]=(v_{ij})$, where

 $v_{ij}=\begin{cases}N\pi_{i}(1-\pi_{i})&\text{if i=j;}\\ -N\pi_{i}\pi_{j}&\text{if i\neq j.}\end{cases}$
• When $n=2$, the multinomial distribution is the same as the binomial distribution

• If $X_{1},\ldots,X_{n}$ are mutually independent Poisson random variables with parameters $\lambda_{1},\ldots,\lambda_{n}$ respectively, then the conditional joint distribution of $X_{1},\ldots,X_{n}$ given that $X_{1}+\cdots+X_{n}=N$ is multinomial with parameters $\lambda_{i}/\lambda$, where $\lambda=\sum\lambda_{i}$.

Sketch of proof. Each $X_{i}$ is distributed as:

 $f_{X_{i}}(x_{i})=\frac{e^{-\lambda_{i}}\lambda_{i}^{x_{i}}}{x_{i}!}$

The mutual independence of the $X_{i}$’s shows that the joint probability distribution of the $X_{i}$’s is given by

 $f_{\textbf{X}}(\boldsymbol{x})=\prod_{i=1}^{n}\frac{e^{-\lambda_{i}}\lambda_{i% }^{x_{i}}}{x_{i}!}=e^{-\lambda}\prod_{i=1}^{n}\frac{\lambda_{i}^{x_{i}}}{x_{i}% !},$

where $\textbf{X}=(X_{1},\ldots,X_{n})$, $\boldsymbol{x}=(x_{1},\ldots,x_{n})$ and $\lambda=\lambda_{1}+\cdots+\lambda_{n}$. Next, let $X=X_{1}+\cdots+X_{n}$. Then $X$ is Poisson distributed with parameter $\lambda$ (which can be shown by using induction and the mutual independence of the $X_{i}$’s):

 $f_{X}(x)=\frac{e^{-\lambda}\lambda^{x}}{x!}.$

The conditional probability distribution of X given that $X=N$ is thus given by:

 $f_{\textbf{X}}(\boldsymbol{x}\mid X=N)=\frac{f_{\textbf{X}}(\boldsymbol{x})}{f% _{X}(N)}=(e^{-\lambda}\prod_{i=1}^{n}\frac{\lambda_{i}^{x_{i}}}{x_{i}!})/(% \frac{e^{-\lambda}\lambda^{N}}{N!})=\frac{N!}{x_{1}!\cdots x_{n}!}\prod_{i=1}^% {n}(\frac{\lambda_{i}}{\lambda})^{x_{i}},$

where $\sum x_{i}=N$ and that $\sum\lambda_{i}/\lambda=1$.

Title multinomial distribution MultinomialDistribution 2013-03-22 14:33:35 2013-03-22 14:33:35 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 60E05