This example shows how to generate random numbers, compute and plot the pdf, and compute descriptive statistics of a multinomial distribution using probability distribution objects.

Create a vector `p`

containing the probability of each outcome. Outcome 1 has a probability of 1/2, outcome 2 has a probability of 1/3, and outcome 3 has a probability of 1/6. The number of trials `n`

in each experiment is 5, and the number of repetitions `reps`

of the experiment is 8.

p = [1/2 1/3 1/6]; n = 5; reps = 8;

Create a multinomial probability distribution object using the specified value `p`

for the `Probabilities`

parameter.

pd = makedist('Multinomial','Probabilities',p)

pd = MultinomialDistribution Probabilities: 0.5000 0.3333 0.1667

Generate one random number from the multinomial distribution, which is the outcome of a single trial.

rng('default') % For reproducibility r = random(pd)

r = 2

This trial resulted in outcome 2.

You can also generate a matrix of random numbers from the multinomial distribution, which reports the results of multiple experiments that each contain multiple trials. Generate a matrix that contains the outcomes of an experiment with `n = 5`

trials and `reps = 8`

repetitions.

r = random(pd,reps,n)

`r = `*8×5*
3 3 3 2 1
1 1 2 2 1
3 3 3 1 2
2 3 2 2 2
1 1 1 1 1
1 2 3 2 3
2 1 3 1 1
3 1 2 1 1

Each element in the resulting matrix is the outcome of one trial. The columns correspond to the five trials in each experiment, and the rows correspond to the eight experiments. For example, in the first experiment (corresponding to the first row), one of the five trials resulted in outcome 1, one of the five trials resulted in outcome 2, and three of the five trials resulted in outcome 3.

Compute the pdf of the distribution.

x = 1:3; y = pdf(pd,x); bar(x,y) xlabel('Outcome') ylabel('Probability Mass') title('Trinomial Distribution')

The plot shows the probability mass for each $$k$$ possible outcome. For this distribution, the pdf value for any `x`

other than 1, 2, or 3 is 0.

Compute the mean, median, and standard deviation of the distribution.

m = mean(pd)

m = 1.6667

med = median(pd)

med = 1

s = std(pd)

s = 0.7454