parameters = number of trials (integer)support =
event probabilities ()|
cdf =|mean =|
mode =|variance =
entropy =|mgf =|
char =|conjugate =Dirichlet: | }} In probability theory, the multinomial distribution is a generalization of the binomial distribution.
The binomial distribution is the probability distribution of the number of "successes" in n independent Bernoulli trials, with the same probability of "success" on each trial. In a multinomial distribution, each trial results in exactly one of some fixed finite number k of possible outcomes, with probabilities p1, ..., pk (so that pi ≥ 0 for i = 1, ..., k and ), and there are n independent trials. Then let the random variables indicate the number of times outcome number i was observed over the n trials. follows a multinomial distribution with parameters n and p, where p = (p1, ..., pk).
for non-negative integers x1, ..., xk.
The expected value of draws in the ith bin is
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 nonnegative-definite matrix of rank k − 1.
The off-diagonal 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
First, reorder the parameters such that they are sorted descendingly (this is only to speed up computation and not strictly necessary). Now, for each trial, draw an auxiliary variable from a uniform distribution. The resulting outcome is the component .
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