Properties
The expected number of times the outcome i was observed over n trials is
The covariance matrix is as follows. Each diagonal entry is the variance of a binomially distributed random variable, and is therefore
The off-diagonal entries are the covariances:
for i, j distinct.
All covariances are negative because for fixed n, an increase in one component of a multinomial vector requires a decrease in another component.
This is a k × k positive-semidefinite matrix of rank k − 1. In the special case where k = n and where the pi are all equal, the covariance matrix is the centering matrix.
The entries of the corresponding correlation matrix are
Note that the sample size drops out of this expression.
Each of the k components separately has a binomial distribution with parameters n and pi, for the appropriate value of the subscript i.
The support of the multinomial distribution is the set
Its number of elements is
the number of n-combinations of a multiset with k types, or multiset coefficient.
Read more about this topic: Multinomial Distribution
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—Ralph Waldo Emerson (1803–1882)
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—John Locke (1632–1704)