Multinomial Distribution

Let a set of random variates , , ..., have a probability function

$\begin{displaymath} P(X_1=x_1, \ldots, X_n=x_n)={N!\over\prod_{i=1}^n x_i!} \prod_{i=1}^n{\theta_i}^{x_i} \end{displaymath}$

(1)

where

are Positive Integers, $\theta_i>0$ , and

$\begin{displaymath} \sum_{i=1}^n \theta_i=1 \end{displaymath}$

(2)

$\begin{displaymath} \sum_{i=1}^n x_i=N. \end{displaymath}$

(3)

Then the joint distribution of

, ...,

is a multinomial distribution and $P(X_1=x_1, \ldots, X_n=x_n)$ is given by the corresponding coefficient of the Multinomial Series

$\begin{displaymath} (\theta_1+\theta_2+\ldots+\theta_n)^N. \end{displaymath}$

(4)

The Mean and Variance of

are

$\displaystyle \mu_i$	$\textstyle =$	$\displaystyle N\theta_i$	(5)
$\displaystyle {\sigma_i}^2$	$\textstyle =$	$\displaystyle N\theta_i(1-\theta_i).$	(6)

and

$\begin{displaymath} {\sigma_{ij}}^2=-N\theta_i\theta_j. \end{displaymath}$

(7)

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 532, 1987.