 Multivariate normal distribution

A random vector $X\in {\mathbb{R}}^d$ has a multivariate normal distribution when every real linear combination of its coordinates is normal, allowing zero variance. Its mean $\mu$ and covariance matrix $\Sigma$ determine the characteristic functionIf $\Sigma$ is positive definite, its probability density function is $(2\pi {)}^{-d/2}(\det \Sigma {)}^{-1/2}\exp [-(x-\mu {)}^T{\Sigma }^{-1}(x-\mu )/2]$. A singular covariance matrix gives a law supported on a proper affine subspace, without such a full-dimensional density. Joint normality makes zero cross-covariance equivalent to independence, unlike general distributions.