Probability mass function:

Cumulative distribution function:

Mean and variance:

Covariance:

Generate a sample of pseudorandom vectors from a negative multinomial distribution:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Goodness-of-fit test:

Skewness:

Kurtosis:

Correlation:

Different mixed moments for a bivariate negative multinomial distribution:

Mixed central moments:

Mixed factorial moments:

Closed form for a symbolic order:

Mixed cumulants:

Hazard function:

Univariate marginals give NegativeBinomialDistribution:

Multivariate marginals give NegativeMultinomialDistribution:

Use dimensionless Quantity to define NegativeMultinomialDistribution:

Find the distribution for the number of times a biased coin should be flipped until you get heads twice in a row. If you let p be the probability of heads, the two events for which you start over are "T" (tail) or "HT" (head, tail) and the event for which you succeed is "HH" (head, head). These events have the following probabilities:

The total number of coin flips until two heads:

Find the probability that no more than five coin flips are required:

The components are correlated:

Relationships to other distributions:

A univariate negative multinomial distribution is a negative binomial distribution:

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

NegativeMultinomialDistribution is not defined when p is not a vector of probabilities that sum to less than 1:

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