Page 252 - Elements of Distribution Theory

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238 Normal Distribution Theory

1
Corollary 3.2 that X has the same distribution as µ + 2 Z. Hence, part (vii) of the theorem
holds.

Example 8.1 (Bivariate normal distribution). Suppose that X = (X 1 , X 2 ) has a two-
dimensional multivariate normal distribution. Then , the covariance matrix of the distri-
bution, is of the form
2
σ 1 σ 12
= 2
σ 21 σ
2
2
where σ denotes the variance of X j , j = 1, 2, and σ 12 = σ 21 denotes the covariance of
j
X i , X j .
We may write σ 12 = ρσ 1 σ 2 so that ρ denotes the correlation of X 1 and X 2 . Then

σ 1 0 1 ρ σ 1 0
= .
0 σ 2 ρ 1 0 σ 2
It follows that is nonnegative-definite provided that

1 ρ
ρ 1
is nonnegative-definite. Since this matrix has eigenvalues 1 − ρ, 1 + ρ, is nonnegative-
definite for any −1 ≤ ρ ≤ 1. If ρ =±1, then X 1 /σ 1 − ρX 2 /σ 2 has variance 0.

Example 8.2 (Exchangeable normal random variables). Consider a multivariate normal
random vector X = (X 1 , X 2 ,..., X n ) and suppose that X 1 , X 2 ,..., X n are exchangeable
random variables. Let µ and denote the mean vector and covariance matrix, respectively.
Then, according to Theorem 2.8, each X j has the same marginal distribution; hence, µ
must be a constant vector and the diagonal elements of must be equal. Also, each pair
(X i , X j ) must have the same distribution; it follows that the Cov(X i , X j )isa constant, not
depending on i, j. Hence, must be of the form
1 ρρ ··· ρ
 
 ρ 1 ρ ··· ρ 
= σ 2  . 
.
 . 
ρρρ ··· 1
for some constants σ ≥ 0 and ρ.Of course, is a valid covariance matrix only for certain
values of ρ; see Exercise 8.8.

Example 8.3 (Principal components). Let X denote a d-dimensional random vector with
a multivariate normal distribution with mean vector µ and covariance matrix . Consider
T
the problem of finding the linear function a X with the maximum variance; of course, the
T
variance of a X can be made large by choosing the elements of a to be large in magnitude.
Hence, we require a to be a unit vector. Since
T
T
Var(a X) = a a,
T
we want to find the unit vector a that maximizes a a.