Introduction to Probability, Second Edition, 2nd Edition

7.5    Multivariate Normal

The Multivariate Normal is a continuous multivariate distribution that generalizes the Normal distribution into higher dimensions. We will not work with the rather unwieldy joint PDF of the Multivariate Normal. Instead we define the Multivariate Normal by its relationship to the ordinary Normal.

Definition 7.5.1 (Multivariate Normal distribution). A k-dimensional random vector $X=\left(X_1,\dots ,X_k\right)$ is said to have a Multivariate Normal (MVN) distribution if every linear combination of the $X_j$ has a Normal distribution. That is, we require

$t_1{X}_1+\cdots +t_k{X}_k$

to have a Normal distribution for any constants $t_1,\dots ,t_k$. If $t_1{X}_1+\cdots +t_k{X}_k$ is a constant (such as when all $t_i=0$), we consider it to have a Normal distribution, albeit a degenerate ...