Page 249 - Elements of Distribution Theory

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8

Normal Distribution Theory

8.1 Introduction
The normal distribution plays a central role in statistical theory and practice, both as a model
for observed data and as a large-sample approximation to the distribution of wide range of
statistics, as will be discussed in Chapters 11–13. In this chapter, we consider in detail the
distribution theory associated with the normal distribution.

8.2 Multivariate Normal Distribution
A d-dimensional random vector X has a multivariate normal distribution with mean vector
d
d
T
µ ∈ R and covariance matrix if, for any a ∈ R , a X has a normal distribution with
T
T
mean a µ and variance a a. Here is a d × d nonnegative-definite, symmetric matrix.
T
T
T
Note that a a might be 0, in which case a X = a µ with probability 1.
The following result establishes several basic properties of the multivariate normal dis-
tribution.
Theorem 8.1. Let X be a d-dimensional random vector with a multivariate normal distri-
bution with mean vector µ and covariance matrix .
(i) The characteristic function of X is given by
1 T d

T
ϕ(t) = exp it µ − t t , t ∈ R .
2
(ii) Let B denote a p × d matrix. Then BX has a p-dimensional multivariate normal
T
distribution with mean vector Bµ and covariance matrix B B .
(iii) Suppose that the rank of is r < d. Then there exists a (d − r)-dimensional
d
subspace of R ,V , such that for any v ∈ V,
T
Pr{v (X − µ) = 0}= 1.
There exists an r × d matrix C such that Y = CX has a multivariate normal
distribution with mean Cµ and diagonal covariance matrix of full rank.
(iv) Let X = (X 1 , X 2 ) where X 1 is p-dimensional and X 2 is (d − p)-dimensional. Write
p
µ = (µ 1 ,µ 2 ) where µ 1 ∈ R and µ 2 ∈ R d−p , and write

11 12
=
21 22
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