## Wishart Distribution

### Overview

The Wishart distribution is a generalization of the univariate
chi-square distribution to two or more variables. It is a distribution
for symmetric positive semidefinite matrices, typically covariance
matrices, the diagonal elements of which are each chi-square random
variables. In the same way as the chi-square distribution can be constructed
by summing the squares of independent, identically distributed, zero-mean
univariate normal random variables, the Wishart distribution can be
constructed by summing the inner products of independent, identically
distributed, zero-mean multivariate normal random vectors. The Wishart
distribution is often used as a model for the distribution of the
sample covariance matrix for multivariate normal random data, after
scaling by the sample size.

Only random matrix generation is supported for the Wishart distribution,
including both singular and nonsingular Σ.

### Parameters

The Wishart distribution is parameterized with a symmetric,
positive semidefinite matrix, Σ, and a positive scalar degrees
of freedom parameter, ν. ν is analogous to the degrees
of freedom parameter of a univariate chi-square distribution, and
Σν is the mean of the distribution.

### Probability Density Function

The probability density function of the *d*-dimensional
Wishart distribution is given by

$$\text{y=f(}{\rm X}\text{,}\Sigma \text{,}\nu \text{)=}\frac{{\left|{\rm X}\right|}^{\left(\text{(}\nu \text{-d-1})/2\right)}{e}^{\left(\text{-}\frac{1}{2}\text{trace}\left({\Sigma}^{-1}{\rm X}\right)\right)}}{{\text{2}}^{\text{(}\nu \text{d)/2}}{\pi}^{\text{(d(d-1))/4}}{|\sum |}^{\nu /2}\Gamma \left(\nu /2\right)\mathrm{...}\Gamma \text{(}\nu \text{-(d-1))/2}}$$

where *X* and Σ are *d*-by-*d* symmetric
positive definite matrices, and ν is a scalar greater than *d* –
1. While it is possible to define the Wishart for singular Σ,
the density cannot be written as above.

### Example

If *x* is a bivariate normal random vector
with mean zero and covariance matrix

$$\Sigma =\left(\begin{array}{cc}1& .5\\ .5& 2\end{array}\right)$$

then you can use the Wishart distribution to generate a sample
covariance matrix without explicitly generating *x* itself.
Notice how the sampling variability is quite large when the degrees
of freedom is small.

Sigma = [1 .5; .5 2];
df = 10; S1 = wishrnd(Sigma,df)/df
S1 =
1.7959 0.64107
0.64107 1.5496
df = 1000; S2 = wishrnd(Sigma,df)/df
S2 =
0.9842 0.50158
0.50158 2.1682

## See Also

`wishrnd`

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