Multivariate Normal Distributions

Abstract

In this lecture we consider a common example of a multivariate distribution. We begin with the non-degenerate case. Let ξ₁..., ξ _n be n random variables whose joint distribution is given by the density p(x ₁,...,x _n ), of the form % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI % cacaWG4bGaaiykaiabg2da9iaadchacaGGOaGaamiEamaaBaaaleaa % caaIXaaabeaakiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamiEam % aaBaaaleaacaWGUbaabeaakiaacMcacqGH9aqpcaWGdbGaamyzamaa % CaaaleqabaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOa % GaamyqaiaacIcacaWG4bGaeyOeI0IaamyBaiaacMcacaGGSaGaaiik % aiaadIhacqGHsislcaWGTbGaaiykaiaacMcaaaGccaGGUaaaaa!556B! \[p(x) = p({x_1},...,{x_n}) = C{e^{ - \frac{1}{2}(A(x - m),(x - m))}}.\] Here C is a normalizing constant, x = (x ₁,..., x _n ) is an n—dimensional vector, m = (m ₁ , ... , m _n ) is also an n-dimensional vector, and A is a symmetric matrix. The density (9.1) is said to be the density of a non-degenerate multivariate normal distribution. The convenience of (9.1) lies in the fact that the density is defined in terms of very simple parameters: an n—dimensional vector m and a symmetric matrix A of order n.