Multivariate t-distribution

Multivariate t

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

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.

One common method of construction of a multivariate t-distribution, for the case of ${\displaystyle p}$ dimensions, is based on the observation that if ${\displaystyle \mathbf {y} }$ and ${\displaystyle u}$ are independent and distributed as ${\displaystyle {\mathcal {N}}({\mathbf {0} },{\boldsymbol {\Sigma }})}$ and ${\displaystyle \chi _{\nu }^{2}}$ (i.e. multivariate normal and chi-squared distributions) respectively, the covariance ${\displaystyle \mathbf {\Sigma } \,}$ is a p × p matrix, and ${\displaystyle {\mathbf {y} }/{\sqrt {u/\nu }}={\mathbf {x} }-{\boldsymbol {\mu }}}$, then ${\displaystyle {\mathbf {x} }}$ has the density

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