# Wishart distribution

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 ${\displaystyle \Theta \mapsto \left|{\mathbf {I} }-2i\,{\mathbf {\Theta } }{\mathbf {V} }\right|^{-{\frac {n}{2}}}}$