Function Repository Resource:

WishartDistribution

Source Notebook

Represent the Wishart distribution

Contributed by: Wolfram Research

ResourceFunction["WishartDistribution"][Σ,m]

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

Details and Options

The probability density for a symmetric positive definite matrix x 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.

ResourceFunction["WishartDistribution"] can be used with such functions as Mean, PDF and RandomReal.

Examples

Basic Examples (3)

The mean of a Wishart distribution:

In[1]:=

$Mean[ResourceFunction[ "WishartDistribution"][{{Subscript[\[Sigma], 11]^2, \[Rho]*Subscript[\[Sigma], 11]*Subscript[\[Sigma], 22]}, {\[Rho]*Subscript[\[Sigma], 11]*Subscript[\[Sigma], 22], Subscript[\[Sigma], 22]^2}}, 10]]$

Out[1]=

The variance:

In[2]:=

$Variance[ ResourceFunction[ "WishartDistribution"][{{Subscript[\[Sigma], 11]^2, \[Rho]*Subscript[\[Sigma], 11]*Subscript[\[Sigma], 22]}, {\[Rho]*Subscript[\[Sigma], 11]*Subscript[\[Sigma], 22], Subscript[\[Sigma], 22]^2}}, 10]]$

Out[2]=

Probability density function:

In[3]:=

$PDF[ResourceFunction[ "WishartDistribution"][{{1, \[Rho]}, {\[Rho], 1}}, 10], {{Subscript[x, 11], Subscript[x, 12]}, {Subscript[x, 12], Subscript[x, 22]}}]$