Multivariate t-distribution

Multivariate t
Notation	$t_{p}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }},\nu )$
Parameters	${\boldsymbol {\mu }}=[\mu _{1},\dots ,\mu _{p}]^{\mathsf {T}}$ location (real $p\times 1$ vector) ${\boldsymbol {\Sigma }}$ scale matrix (positive-definite real $p\times p$ matrix) $\nu >0$ (real) represents the degrees of freedom
Support	$\mathbf {x} \in \mathbb {R} ^{p}\!$
PDF	${\frac {\Gamma \left[(\nu +p)/2\right]}{\Gamma (\nu /2)\nu ^{p/2}\pi ^{p/2}\left|{\boldsymbol {\Sigma }}\right|^{1/2}}}\left[1+{\frac {1}{\nu }}({\mathbf {x} }-{\boldsymbol {\mu }})^{\mathsf {T}}{\boldsymbol {\Sigma }}^{-1}({\mathbf {x} }-{\boldsymbol {\mu }})\right]^{-(\nu +p)/2}$
CDF	No analytic expression, but see text for approximations
Mean	${\boldsymbol {\mu }}$ if $\nu >1$; else undefined
Median	${\boldsymbol {\mu }}$
Mode	${\boldsymbol {\mu }}$
Variance	${\frac {\nu }{\nu -2}}{\boldsymbol {\Sigma }}$ (covariance matrix) if $\nu >2$; else undefined
Skewness	0 if $\nu >3$; else undefined

In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's t-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix t-distribution is distinct and makes particular use of the matrix structure.