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


  • The probability for a vector of non-negative integers , , , , where is the length of in a negative multinomial distribution, is proportional to .
  • 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 allows n and components of vector p to be dimensionless quantities. »
  • NegativeMultinomialDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

  • NegativeMultinomialDistribution[n,{p1,p2,,pm}] represents a discrete multivariate statistical distribution supported over the subset of consisting of all tuples of integers satisfying and characterized by the property that each of the ^(th) (univariate) marginal distributions has a NegativeBinomialDistribution for . In other words, each of the variables satisfies xjBinomialDistribution[n,qj] for some probabilities qj where . The negative multinomial distribution is parametrized by a positive real number n and a vector {p1,p2,,pm} of non-negative real numbers satisfying (called a "failure probability vector"), which together define the associated mean, variance, and covariance of the distribution.
  • The negative multinomial distribution was first investigated in the mid-1950s as a multivariate analogue of the univariate negative binomial distribution (NegativeBinomialDistribution). Analogously to the univariate case, NegativeMultinomialDistribution[n,{p1,p2,,pm}] for integer n gives the distribution of the failure counts in a sequence of trials with success probability and m types of failure before n successes occur. The negative multinomial distribution has been used to model phenomena including epidemics, accident frequency, reliability, and industrial absenteeism.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a negative multinomial distribution. Distributed[x,NegativeMultinomialDistribution[n,{p1,p2,,pm}]] , written more concisely as xNegativeMultinomialDistribution[n,{p1,p2,,pm}], can be used to assert that a random variable x is distributed according to a negative multinomial distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions for negative multinomial distributions may be given using PDF[NegativeMultinomialDistribution[n,{p1,p2,,pm}]] and CDF[NegativeMultinomialDistribution[n,{p1,p2,,pm}]]. The mean, median, variance, covariance, raw moments, and central moments may be computed using Mean, Median, Variance, Covariance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a negative multinomial distribution, EstimatedDistribution to estimate a negative multinomial parametric distribution from given data, and FindDistributionParameters to fit data to a negative multinomial distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic negative multinomial distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic negative multinomial distribution.
  • TransformedDistribution can be used to represent a transformed negative multinomial distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a negative multinomial distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving negative multinomial distributions.
  • NegativeMultinomialDistribution is related to a number of other distributions. It is a higher-dimensional generalization of NegativeBinomialDistribution in the sense that the PDF of NegativeBinomialDistribution[n,p] with respect to a variable x is precisely the same as that of NegativeMultinomialDistribution[n,{1-p}] written in terms of the vector {x}. NegativeMultinomialDistribution is also related to BinomialDistribution, MultinomialDistribution, BernoulliDistribution, BetaBinomialDistribution, HypergeometricDistribution, GeometricDistribution, and PoissonDistribution.


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Basic Examples  (4)

Probability mass function:

Cumulative distribution function:

Mean and variance:


Scope  (9)

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:




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:

Applications  (1)

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:

Properties & Relations  (3)

The components are correlated:

Relationships to other distributions:

A univariate negative multinomial distribution is a negative binomial distribution:

Possible Issues  (3)

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:

Wolfram Research (2010), NegativeMultinomialDistribution, Wolfram Language function, (updated 2016).


Wolfram Research (2010), NegativeMultinomialDistribution, Wolfram Language function, (updated 2016).


Wolfram Language. 2010. "NegativeMultinomialDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016.


Wolfram Language. (2010). NegativeMultinomialDistribution. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_negativemultinomialdistribution, author="Wolfram Research", title="{NegativeMultinomialDistribution}", year="2016", howpublished="\url{}", note=[Accessed: 15-July-2024 ]}


@online{reference.wolfram_2024_negativemultinomialdistribution, organization={Wolfram Research}, title={NegativeMultinomialDistribution}, year={2016}, url={}, note=[Accessed: 15-July-2024 ]}