Multivariate normal distribution

Remarks

Here $\mu$ is the mean vector, with $\mu_i = E[X_i]$, and $\Sigma$ is the covariance matrix, with $\Sigma_{ij} = \operatorname{Cov}(X_i, X_j)$. An equivalent condition: $X$ is multivariate normal if and only if $a^{\mathsf{T}} X$ has a normal distribution for each $a \in \mathbb{R}^k$. Here a constant counts as normal with variance $0$. If $\Sigma$ is positive definite, $X$ has the density $$f(x) = \frac{1}{\sqrt{(2\pi)^k \det \Sigma}} \exp\left(-\tfrac{1}{2} (x - \mu)^{\mathsf{T}} \Sigma^{-1} (x - \mu)\right), \quad x \in \mathbb{R}^k.$$ The components of a multivariate normal vector are independent if and only if they are uncorrelated, that is, if $\Sigma$ is diagonal.