Introduction to Probability and Stochastic Processes with Applications

CHAPTER 7

MULTIVARIATE NORMAL DISTRIBUTIONS

The multivariate normal distributions is one of the most important multidimensional distributions and is essential to multivariate statistics. The multivariate normal distribution is an extension of the univariate normal distribution and shares many of its features. This distribution can be completely described by its means, variances and covariances given in this chapter. The brief introduction to this distribution given will be necessary for students who wish to take the next course in multivariate statistics but can be skipped otherwise.

7.1   MULTIVARIATE NORMAL DISTRIBUTION

Definition 7.1 (Multivariate Normal Distribution) An n-dimensional random vector X = (X₁, …, X_n) is said to have a multivariate normal distribution if any linear combination has a univariate normal distribution (possibly degenerated, as happens, for example, when α_j = 0 for all j).

  EXAMPLE 7.1

Suppose that X₁, …, X_n are n independent random variables such that

for j = 1, …, n. Then, if , we have

where

In other words,

Therefore, the vector X : = (X₁, …, X_n) has ...