A multinomial distribution arises when there are \(n\) independent "trials", each with the same probabilities for the \(g\) different possible values from each trial. We now concentrate on the number of times that one of these values is observed — its marginal distribution.

Marginal distributions

If \((X_1, X_2,\dots, X_g)\), have a \(\MultinomDistn(n, \pi_1, \dots, \pi_g)\) distribution, then the marginal distribution of any \(X_i\) is \(BinomDistn(n, \pi_i)\).

This follows from treating the outcome \(O_i\) as a "success" and combining all other outcomes as a "failure". \(X_i\) is then the number of successes in \(n\) independent trials, each with probability \(\pi_i\) of success.

 

Opinion poll

We return here to the example on the previous page where responses to a question about a new piece of legislation had probabilities

P(Agree) = 0.3,
P(Neutral) = 0.4
P(Disagree) = 0.3

The 3-dimensional bar chart below again shows the joint probabilities for the numbers in a random sample of \(n\) who agree and are neutral.

Initially the sample size is \(n = 1\). Click the box Marginal for X to display the marginal probabilities for the number agreeing. Since the probability of agreeing is 0.3, the marginal distribution is a \(\BinomDistn(n=1, \pi=0.3)\) distribution.

Similarly the marginal distribution for the number who are neutral can be displayed by clicking Marginal for Y and is \(\BinomDistn(n=1, \pi=0.4)\).

Increase the sample size to display the marginal binomial distributions for other sample sizes.