MultivariateStatistics`
MultivariateStatistics`

# NegativeMultinomialDistribution

As of Version 8, NegativeMultinomialDistribution is part of the built-in Wolfram Language kernel.

NegativeMultinomialDistribution[n,p]

represents a negative multinomial distribution with parameter n and failure probability vector p.

# Details and Options

• To use NegativeMultinomialDistribution, you first need to load the Multivariate Statistics Package using Needs["MultivariateStatistics`"].
• The probability for a vector x of non-negative integers x1, x2, , xLength[p] in a negative multinomial distribution is proportional to (n-1+xi)!(pixi/xi!).
• The parameter n can be any positive real number, and p can be any vector of non-negative real numbers that sum to less than unity.
• If n is a positive integer, NegativeMultinomialDistribution[n,p] gives the distribution of the failure counts in a sequence of trials with success probability 1-Total[p] and Length[p] types of failure before n successes occur.
• NegativeMultinomialDistribution can be used with such functions as Mean, CDF, and RandomInteger.

# Examples

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## Basic Examples(3)

The mean of a negative multinomial distribution:

The variances of each dimension:

Probability density function:

## Scope(3)

Generate a set of pseudorandom vectors that follow a negative multinomial distribution:

## Properties & Relations(1)

A univariate negative multinomial distribution is a negative binomial distribution:

## Possible Issues(2)

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

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

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

Wolfram Research (2007), NegativeMultinomialDistribution, Wolfram Language function, https://reference.wolfram.com/language/MultivariateStatistics/ref/NegativeMultinomialDistribution.html.

#### Text

Wolfram Research (2007), NegativeMultinomialDistribution, Wolfram Language function, https://reference.wolfram.com/language/MultivariateStatistics/ref/NegativeMultinomialDistribution.html.

#### CMS

Wolfram Language. 2007. "NegativeMultinomialDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/MultivariateStatistics/ref/NegativeMultinomialDistribution.html.

#### APA

Wolfram Language. (2007). NegativeMultinomialDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/MultivariateStatistics/ref/NegativeMultinomialDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_negativemultinomialdistribution, author="Wolfram Research", title="{NegativeMultinomialDistribution}", year="2007", howpublished="\url{https://reference.wolfram.com/language/MultivariateStatistics/ref/NegativeMultinomialDistribution.html}", note=[Accessed: 19-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_negativemultinomialdistribution, organization={Wolfram Research}, title={NegativeMultinomialDistribution}, year={2007}, url={https://reference.wolfram.com/language/MultivariateStatistics/ref/NegativeMultinomialDistribution.html}, note=[Accessed: 19-June-2024 ]}