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MultiPoissonDistribution[μ0,μ]

represents a multivariate Poisson distribution with mean vector μ0+μ.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Possible Issues  
See Also
Tech Notes
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MultivariateStatistics`
MultivariateStatistics`

MultiPoissonDistribution

As of Version 8.0, MultiPoissonDistribution has been renamed to MultivariatePoissonDistribution and is part of the built-in Wolfram Language kernel.

MultiPoissonDistribution[μ0,μ]

represents a multivariate Poisson distribution with mean vector μ0+μ.

Details and Options

  • To use MultiPoissonDistribution, you first need to load the Multivariate Statistics Package using Needs["MultivariateStatistics`"].
  • The multivariate Poisson distribution MultiPoissonDistribution[μ0,μ] with μ={μ1,μ2,…} is the distribution followed by a Poisson with mean μ0 summed with a vector of independent Poissons with means μ1, μ2, ….
  • The parameter μ0 and the elements of the vector μ can be any positive numbers.
  • MultiPoissonDistribution can be used with such functions as Mean, PDF, and RandomInteger.

Examples

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Basic Examples  (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

The mean of a multivariate Poisson distribution:

Wolfram Language code: Mean[MultiPoissonDistribution[Subscript[μ, 0], {Subscript[μ, 1], Subscript[μ, 2]}]]
Wolfram Language code: Needs["MultivariateStatistics`"]

The variances of each dimension:

Wolfram Language code: Variance[MultiPoissonDistribution[Subscript[μ, 0], {Subscript[μ, 1], Subscript[μ, 2]}]]
Wolfram Language code: Needs["MultivariateStatistics`"]

Probability density function:

Wolfram Language code: PDF[MultiPoissonDistribution[Subscript[μ, 0], {Subscript[μ, 1], Subscript[μ, 2]}], {x, y}]

Scope  (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

Generate a set of pseudorandom vectors that follow a multivariate Poisson distribution:

Wolfram Language code: RandomInteger[MultiPoissonDistribution[5, {1, 2, 10}], 10]
Wolfram Language code: Needs["MultivariateStatistics`"]
Wolfram Language code: Skewness[MultiPoissonDistribution[Subscript[μ, 0], {Subscript[μ, 1], Subscript[μ, 2]}]]
Wolfram Language code: Needs["MultivariateStatistics`"]
Wolfram Language code: Kurtosis[MultiPoissonDistribution[Subscript[μ, 0], {Subscript[μ, 1], Subscript[μ, 2]}]]

Possible Issues  (2)

Wolfram Language code: Needs["MultivariateStatistics`"]

MultiPoissonDistribution is not defined when μ0 is not positive:

Wolfram Language code: Mean[MultiPoissonDistribution[-3, {Subscript[μ, 1], Subscript[μ, 2]}]]

MultiPoissonDistribution is not defined when any of the elements of μ are not positive:

Wolfram Language code: Mean[MultiPoissonDistribution[Subscript[μ, 0], {-2, Subscript[μ, 2]}]]
Wolfram Language code: Needs["MultivariateStatistics`"]

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

Wolfram Language code: Mean[MultiPoissonDistribution[Subscript[μ, 0], {Subscript[μ, 1], Subscript[μ, 2]}]] /. {Subscript[μ, 0] -> -5, Subscript[μ, 1] -> 1 / 2, Subscript[μ, 2] -> I}
Wolfram Research (2007), MultiPoissonDistribution, Wolfram Language function, https://reference.wolfram.com/language/MultivariateStatistics/ref/MultiPoissonDistribution.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_multipoissondistribution, author="Wolfram Research", title="{MultiPoissonDistribution}", year="2007", howpublished="\url{https://reference.wolfram.com/language/MultivariateStatistics/ref/MultiPoissonDistribution.html}", note=[Accessed: 17-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_multipoissondistribution, organization={Wolfram Research}, title={MultiPoissonDistribution}, year={2007}, url={https://reference.wolfram.com/language/MultivariateStatistics/ref/MultiPoissonDistribution.html}, note=[Accessed: 17-June-2026]}