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WishartDistribution[Σ,m]

represents a Wishart distribution with scale matrix Σ and degrees of freedom parameter m.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Possible Issues  
Tech Notes
Related Guides
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MultivariateStatistics`
MultivariateStatistics`

WishartDistribution

WishartDistribution[Σ,m]

represents a Wishart distribution with scale matrix Σ and degrees of freedom parameter m.

Details and Options

  • To use WishartDistribution, you first need to load the Multivariate Statistics Package using Needs["MultivariateStatistics`"].
  • The probability density for a symmetric positive definite matrix in a Wishart distribution is proportional to .
  • The scale matrix Σ can be any symmetric positive definite matrix. The parameter m can be any number such that m>Length[Σ].
  • For integer m, the Wishart distribution gives the distribution of covariance matrices of multinormal samples.
  • WishartDistribution can be used with such functions as Mean, PDF, and RandomReal.

Examples

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Basic Examples  (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

The mean of a Wishart distribution:

Wolfram Language code: Mean[WishartDistribution[{{Subscript[σ, 11]^2, ρ * Subscript[σ, 11] * Subscript[σ, 22]}, {ρ * Subscript[σ, 11] * Subscript[σ, 22], Subscript[σ, 22]^2}}, 10]]
Wolfram Language code: Needs["MultivariateStatistics`"]

The variance:

Wolfram Language code: Variance[WishartDistribution[{{Subscript[σ, 11]^2, ρ * Subscript[σ, 11] * Subscript[σ, 22]}, {ρ * Subscript[σ, 11] * Subscript[σ, 22], Subscript[σ, 22]^2}}, 10]]
Wolfram Language code: Needs["MultivariateStatistics`"]

Probability density function:

Wolfram Language code: PDF[WishartDistribution[{{1, ρ}, {ρ, 1}}, 10], {{Subscript[x, 11], Subscript[x, 12]}, {Subscript[x, 12], Subscript[x, 22]}}]

Scope  (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

Generate a set of pseudorandom matrices that follow a Wishart distribution:

Wolfram Language code: RandomReal[WishartDistribution[{{1, 1 / 3}, {1 / 3, 1}}, 5], 5]
Wolfram Language code: Needs["MultivariateStatistics`"]
Wolfram Language code: Skewness[WishartDistribution[{{1, 1 / 3}, {1 / 3, 1}}, 5]]
Wolfram Language code: Needs["MultivariateStatistics`"]
Wolfram Language code: Kurtosis[WishartDistribution[{{1, 1 / 3}, {1 / 3, 1}}, 5]]

Possible Issues  (2)

Wolfram Language code: Needs["MultivariateStatistics`"]

WishartDistribution is not defined when Σ is not symmetric and positive definite:

Wolfram Language code: Mean[WishartDistribution[{{1, 2}, {0, 1}}, 5]]

WishartDistribution is not defined when m<Length[Σ]:

Wolfram Language code: Mean[WishartDistribution[{{1, 0}, {0, 1}}, 1]]
Wolfram Language code: Needs["MultivariateStatistics`"]

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

Wolfram Language code: Mean[WishartDistribution[{{Subscript[s, 11], Subscript[s, 12]}, {Subscript[s, 12], Subscript[s, 22]}}, m]] /. {Subscript[s, 11] -> 1, Subscript[s, 22] -> 1, Subscript[s, 12] -> 0, m -> I}
Wolfram Research (2007), WishartDistribution, Wolfram Language function, https://reference.wolfram.com/language/MultivariateStatistics/ref/WishartDistribution.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_wishartdistribution, author="Wolfram Research", title="{WishartDistribution}", year="2007", howpublished="\url{https://reference.wolfram.com/language/MultivariateStatistics/ref/WishartDistribution.html}", note=[Accessed: 15-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_wishartdistribution, organization={Wolfram Research}, title={WishartDistribution}, year={2007}, url={https://reference.wolfram.com/language/MultivariateStatistics/ref/WishartDistribution.html}, note=[Accessed: 15-June-2026]}