Probability mass function:

Cumulative distribution function:

Mean and variance:

Covariance:

Generate a sample of pseudorandom vectors from a multinomial distribution:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Goodness-of-fit test:

Skewness:

Kurtosis:

Correlation:

Hazard function:

The univariate marginals follow BinomialDistribution:

Multivariate marginals do not simplify:

Plot the probability density function of the marginal for fixed values of parameters:

Use dimensionless Quantity to define MultinomialDistribution:

There are two candidates in an election where the winner is chosen based on a simple majority. Each of voters votes for candidate 1 with probability and for candidate 2 with probability , where , so that a voter may choose not to vote for either candidate. When , , the probability of one swing vote is:

Probability that a winner won by one vote:

Probability that candidate 1 wins the election:

Probable outcome of the next election:

Average outcome of an election:

Distribute 5 balls among 3 containers, picking each container with equal probability. Find the probability that no container is empty:

Compute the same probability using SurvivalFunction:

Distribute balls among containers with equal probability. Find the probability that no container is empty for different values of and :

In calling a customer service center, one of three things may happen: the line is busy with probability 0.4, a caller gets the wrong party with probability 0.1, or a caller gets connected to an agent. Find the probability that a caller calling at 6 different times gets a busy signal 4 times and twice connects directly to an agent:

Find the probability that calling at 6 different times, a caller gets the wrong party at least twice:

Simulate 6 calling attempts done at different times:

In a certain city, out of all 911 calls 30% were requesting an ambulance, 15% were requesting the fire department, and the rest were police requests. Find the probability that in the next 10 emergency calls, 2 will ask for an ambulance, 1 for the fire department, and 7 for police:

Simulate the request distribution for the next 100 calls:

Define a multivariate Pólya distribution as a parameter mixture of negative multinomial distribution:

Relationships to other distributions:

A bivariate multinomial distribution is a BinomialDistribution:

A univariate multinomial distribution is a shifted BernoulliDistribution:

MultinomialDistribution is not defined when n is not a positive integer:

MultinomialDistribution is not defined when the elements of p do not sum to 1:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: