Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.5. The Multivariate Normal Distribution 203

whereX 2 is of dimensionp=n−m. In the same way, partition the mean and
covariance matrix ofX;thatis,

μ=

[

μ 1

μ 2

]

andΣ=

[

Σ 11 Σ 12

Σ 21 Σ 22

]

(3.5.19)

with the same dimensions as in expression (3.5.18). Note, for instance, thatΣ 11
is the covariance matrix ofX 1 andΣ 12 contains all the covariances between the
components ofX 1 andX 2. Now defineAto be the matrix

A=[Im

..

.Omp],

whereOmpis anm×pmatrix of zeroes. ThenX 1 =AX. Hence, applying Theorem
3.5.2 to this transformation, along with some matrix algebra, we have the following
corollary:

Corollary 3.5.1.SupposeXhas aNn(μ,Σ)distribution, partitioned as in expres-
sions (3.5.18) and (3.5.19). ThenX 1 has aNm(μ 1 ,Σ 11 )distribution.
This is a useful result because it says that any marginal distribution ofXis also
normal and, further, its mean and covariance matrix are those associated with that
partial vector.
Recall in Section 2.5, Theorem 2.5.2, that if two random variables are indepen-
dent then their covariance is 0. In general, the converse is not true. However, as
the following theorem shows, it is true for the multivariate normal distribution.

Theorem 3.5.3. SupposeXhas aNn(μ,Σ)distribution, partitioned as in the
expressions (3.5.18) and (3.5.19). ThenX 1 andX 2 are independent if and only if
Σ 12 =O.

Proof:First note thatΣ 21 =Σ′ 12. The joint mgf ofX 1 andX 2 is given by

MX 1 ,X 2 (t 1 ,t 2 )=exp

{

t′ 1 μ 1 +t′ 2 μ 2 +

1

2

(t′ 1 Σ 11 t 1 +t′ 2 Σ 22 t 2 +t′ 2 Σ 21 t 1 +t′ 1 Σ 12 t 2 )

}

(3.5.20)
wheret′ =(t′ 1 ,t′ 2 ) is partitioned the same asμ. By Corollary 3.5.1,X 1 has a
Nm(μ 1 ,Σ 11 ) distribution andX 2 has aNp(μ 2 ,Σ 22 ) distribution. Hence, the prod-
uct of their marginal mgfs is

MX 1 (t 1 )MX 2 (t 2 )=exp

{

t′ 1 μ 1 +t′ 2 μ 2 +

1

2

(t′ 1 Σ 11 t 1 +t′ 2 Σ 22 t 2 )

}

. (3.5.21)

By (2.6.6) of Section 2.6,X 1 andX 2 are independent if and only if the expressions
(3.5.20) and (3.5.21) are the same. IfΣ 12 =O′and, hence,Σ 21 =O, then the
expressions are the same andX 1 andX 2 are independent. IfX 1 andX 2 are
independent, then the covariances between their components are all 0; i.e.,Σ 12 =O′
andΣ 21 =O.

Corollary 3.5.1 showed that the marginal distributions of a multivariate normal
are themselves normal. This is true for conditional distributions, too. As the