Page 251 - Elements of Distribution Theory

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052184472Xc08 CUNY148/Severini May 24, 2005 17:54

8.2 Multivariate Normal Distribution 237

1 T d−p
T
ϕ 2 (t) = exp it µ 2 − t 22 t , t ∈ R ,
2
and

1
T T d
ϕ(t) = exp it µ − t t , t ∈ R ;
2
here µ j = E(X j ), j = 1, 2.
p
Let t 1 ∈ R , t 2 ∈ R d−p , and t = (t 1 , t 2 ). Then
T T T
t µ = t µ 1 + t µ 2
1 2
and
T
T
T
T
t t = t 11 t 1 + t 22 t 2 + 2t 12 t 2 .
1 2 1
It follows that
T
ϕ(t) = ϕ 1 (t 1 )ϕ 2 (t 2 )exp −t 12 t 2 .
1
Part (v) of the theorem now follows from Corollary 3.3.
To prove part (vi), let

M 1
M =
M 2
and let Y = MX. Then, by part (ii) of the theorem, Y has a multivariate normal distribution
with covariance matrix
T T
M M 1 M M 2
1 1
T T ;
M M 1 M M 2
2 2
the result now follows from part (v) of the theorem.
d T d
Let a = (a 1 ,..., a d ) ∈ R . Then a Z = a j Z j has characteristic function
j=1

d d d

1 2 2
E exp it a j Z j = E[exp{ita j Z j }] = exp − a t
j
2
j=1 j=1 j=1
d
1
2 2
= exp − a t , t ∈ R,
j
2
j=1
which is the characteristic function of a normal distribution with mean 0 and variance
d 2
j
j=1 a . Hence, Z has a multivariate normal distribution as stated in the theorem.
1 T
Suppose X has the same distribution as µ + 2 Z. Then a X has the same distribution
1
T
T
T
as a µ + a 2 Z, which is normal with mean a µ and variance
1
1
T
T
a 2 2 a = a a;
it follows that X has a multivariate normal distribution with mean vector µ and covariance
matrix .
Now suppose that X has a multivariate normal distribution with mean vector µ and
T
T
d
covariance matrix . Then, for any a ∈ R , a X has a normal distribution with mean a µ
1
T
T
and variance a a. Note that this is the same distribution as a (µ + 2 Z); it follows from