The Multivariate Normal Distribution

The topic of this chapter, which starts with introducing the expectation and the covariance matrix of a d-dimensional random vector, is the general d-variate normal distribution. A d-dimensional random vector X is said to have a d-variate normal distribution if each linear combination of its components has a (possibly degenerate) univariate normal distribution. This definition immediately entails that any collection of components of X has a (lower-dimensional) normal distribution, and that any affine transformation of X has a normal distribution. Moreover, the distribution of X is uniquely determined by the expectation \(\mu \) and the covariance matrix \(\Sigma \) of X. It is shown that for each vector \(\mu \) in \(\mathrm {R}^{\mathrm {d}}\) and each symmetric positive-semidefinite \(d\times d\) matrix \(\Sigma \) , there is a normally distributed random vector X with expectation \(\mu \) and covariance matrix \(\Sigma \) . The chapter proceeds with several properties of the multivariate normal distribution, such as the addition theorem, the density of a non-degenerate d-variate normal distribution, the principal component decomposition, and the chi-square distribution of certain quadratic forms of normally distributed random vectors.