Module Library: RANDMULTINOMIAL Function

RANDMULTINOMIAL Function

generates a random sample from a multinomial distribution

RANDMULTINOMIAL( N, NumTrials, Prob ) ;

The inputs are as follows:

$\text{[math]}$: is the number of desired observations sampled from the distribution.
$\text{[math]}$: is the number of trials for each observation. $\text{[math]}$ , for $\text{[math]}$ .
Prob: is a $\text{[math]}$ vector of probabilities with $\text{[math]}$ and $\text{[math]}$ .

The multinomial distribution is a multivariate generalization of the binomial distribution. For each trial, $\text{[math]}$ is the probability of event $\text{[math]}$ , where the $\text{[math]}$ are mutually exclusive and $\text{[math]}$ .

The RANDMULTINOMIAL function returns an $\text{[math]}$ matrix that contains $\text{[math]}$ observations of $\text{[math]}$ random draws from the multinomial distribution. Each row of the resulting matrix is an integer vector $\text{[math]}$ with $\text{[math]}$ . That is, for each row, $\text{[math]}$ indicates how many times event $\text{[math]}$ occurred in $\text{[math]}$ trials.

If $\text{[math]}$ follows a multinomial distribution with $\text{[math]}$ trials and probabilities $\text{[math]}$ , then

the probability density function for $\text{[math]}$ is

$\text{[math]}$
the expected value of $\text{[math]}$ is $\text{[math]}$ .
the variance of $\text{[math]}$ is $\text{[math]}$ .
the covariance of $\text{[math]}$ with $\text{[math]}$ is $\text{[math]}$ .
if $\text{[math]}$ then $\text{[math]}$ is constant.
if $\text{[math]}$ then $\text{[math]}$ is Binomial $\text{[math]}$ and $\text{[math]}$ is Binomial $\text{[math]}$ .

The following example generates 1000 samples from a multinomial distribution with three mutually exclusive events. For each sample, 10 events are generated. Each row of the returned matrix x represents the number of times each event was observed. The example then computes the sample mean and covariance and compares them with the expected values. Here are the code and the output:

   call randseed(1);
   prob = {0.3,0.6,0.1};
   NumTrials = 10;
   N = 1000;
   x = RANDMULTINOMIAL(N,NumTrials,prob);

   /* population mean and covariance */
   Mean = NumTrials * prob`;
   Cov = -NumTrials*prob*prob`;
   /* replace diagonal elements of Cov with Variance */
   Variance = NumTrials*prob#(1-prob);
   do i = 1 to nrow(prob);
      Cov[i,i] = Variance[i];
   end;

   SampleMean = mean(x);
   SampleCov = cov(x);
   print SampleMean Mean, SampleCov Cov;

            SampleMean                             Mean

     2.971     5.972     1.057              3        6        1

            SampleCov                               Cov

     2.0622212 -1.746559 -0.315663         2.1     -1.8     -0.3
     -1.746559 2.3775936 -0.631035        -1.8      2.4     -0.6
     -0.315663 -0.631035 0.9466977        -0.3     -0.6      0.9

For further details about sampling from the multinomial distribution, see Gentle (2003), or Fishman (1996).