Wishart distribution

Wishart

Notation	X ~ Wp(V, n)
Parameters	n > p − 1 degrees of freedom (real) V > 0 scale matrix (p × p pos. def)
Support	X(p × p) positive definite matrix
PDF	${\displaystyle {\frac {|\mathbf {X} |^{(n-p-1)/2}e^{-\operatorname {tr} (\mathbf {V} ^{-1}\mathbf {X} )/2}}{2^{\frac {np}{2}}|{\mathbf {V} }|^{n/2}\Gamma _{p}({\frac {n}{2}})}}}$ Γp is the multivariate gamma function tr is the trace function
Mean	${\displaystyle \operatorname {E} [X]=nV}$
Mode	(n − p − 1)V for n ≥ p + 1
Variance	${\displaystyle \operatorname {Var} (\mathbf {X} _{ij})=n\left(v_{ij}^{2}+v_{ii}v_{jj}\right)}$
Entropy	see below
CF	${\displaystyle \Theta \mapsto \left|{\mathbf {I} }-2i\,{\mathbf {\Theta } }{\mathbf {V} }\right|^{-{\frac {n}{2}}}}$

${\displaystyle {\frac {|\mathbf {X} |^{(n-p-1)/2}e^{-\operatorname {tr} (\mathbf {V} ^{-1}\mathbf {X} )/2}}{2^{\frac {np}{2}}|{\mathbf {V} }|^{n/2}\Gamma _{p}({\frac {n}{2}})}}}$

In statistics, the Wishart distribution is a generalization to multiple dimensions of the chi-squared distribution, or, in the case of non-integer degrees of freedom, of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.

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