In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.[1]

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Notation X ~ Wp(V, n) n > p − 1 degrees of freedom (real)V > 0 scale matrix (p × p pos. def) X(p × p) positive definite matrix ${\displaystyle f_{\mathbf {X} }(\mathbf {x} )={\frac {|\mathbf {x} |^{(n-p-1)/2}e^{-\operatorname {tr} (\mathbf {V} ^{-1}\mathbf {x} )/2}}{2^{\frac {np}{2}}|{\mathbf {V} }|^{n/2}\Gamma _{p}({\frac {n}{2}})}}}$ Γp is the multivariate gamma function tr is the trace function ${\displaystyle \operatorname {E} [X]=n{\mathbf {V} }}$ (n − p − 1)V for n ≥ p + 1 ${\displaystyle \operatorname {Var} (\mathbf {X} _{ij})=n\left(v_{ij}^{2}+v_{ii}v_{jj}\right)}$ see below ${\displaystyle \Theta \mapsto \left|{\mathbf {I} }-2i\,{\mathbf {\Theta } }{\mathbf {V} }\right|^{-{\frac {n}{2}}}}$
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It is a family of probability distributions defined over symmetric, nonnegative-definite random matrices (i.e. matrix-valued random variables). In random matrix theory, the space of Wishart matrices is called the Wishart ensemble.

These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector.[2]