# Wishart distribution

Let $U_{i}\sim N_{p}(\mu_{i},\Sigma),\quad i=1,\ldots,k$ be independent $p$-dimensional random variables, which are multivariate normally distributed (http://planetmath.org/jointnormaldistribution). Let $S=\sum_{i=1}^{k}U_{i}{U_{i}}^{T}$. Let $M$ be the $k\times p$ matrix with $\mu_{1},\ldots,\mu_{k}$ as rows. Then the joint distribution of the elements of $S$ is said to be a Wishart distribution on $k$ of freedom , and is denoted by $W_{p}(k,\Sigma,M)$. If $M=0$, the distribution is said to be central and is denoted by $W_{p}(k,\Sigma)$. The Wishart distribution is a multivariate generalization of the $\chi^{2}$ distribution.

$W_{p}$ has a density function when $k\geq p$.

Title Wishart distribution WishartDistribution 2013-03-22 16:12:31 2013-03-22 16:12:31 Mathprof (13753) Mathprof (13753) 9 Mathprof (13753) Definition msc 62H05 central Wishart distribution