Probability Distributions and Generating Functions

摘要: The p-dimensional random variable \(X = {[{\textbf{X} }_{1},\ldots,{X}_{p}]}^{\mbox{ T}}\) has a normal distribution, denoted \({\mbox{ N}}_{p}(\boldsymbol\mu,\boldsymbol\Sigma )\), with mean \(\boldsymbol\mu {[{\mu }_{1},\ldots,{\mu }_{p}]}^{\mbox{ and p ×p variance–covariance matrix \(\Sigma \) if its density is of the form $$p(\textbf{x} ) {(2\pi )}^{-p/2}\mid \boldsymbol\Sigma {\mid }^{-1/2} \times \exp \left [-\frac{1} {2}{(x-\mu )}^{\mbox{ T} }{\boldsymbol\Sigma }^{-1}(x-\boldsymbol\mu )\right ],$$ for \(\textbf{} x \in {\mathbb{R}}^{p}\), {\mathbb{R}}^{p}\) non-singular \(\boldsymbol\Sigma \).