Page 225 - Fundamentals of Probability and Statistics for Engineers

208 Fundamentals of Probability and Statistics for Engineers

Theorem 7.4: let X 1 , X 2 ,..., X n be n jointly normally distributed random
variables (not necessarily independent). Then random variable Y , where
Y ˆ c 1 X 1 ‡ c 2 X 2 ‡ ‡ c n X n ; 7:36†
is normally distributed, where c 1 , c 2 , , and c n are constants.
Proof of Theorem 7.4: for convenience, the proof will be given by assuming
that all X j , j ˆ 1, 2, ..., n, have zero means. For this case, the mean of Y is
clearly zero and its variance is, as seen from Equation (4.43),

n n
X X
2
2
ˆ EfY gˆ c i c j ij ; 7:37†
Y
iˆ1 jˆ1
where ˆ cov(X i , X j ).
ij
Since X j are normally distributed, their joint characteristic function is given
by Equation (7.33), which is
!
n n
1 X X
t†ˆ exp ij t i t j : 7:38†
X
2
iˆ1 jˆ1
The characteristic function of Y is
!)
n
X
Y t†ˆ Efexp jtYgˆ E exp jt c k X k
kˆ1
!
n
n
1 2 X X
ˆ exp t ij c i c j
2
iˆ1 jˆ1
1 2 2
ˆ exp t †; 7:39†
Y
2
which is the characteristic function associated with a normal random variable.
Hence Y is also a normal random variable.
A further generalization of the above result is given in Theorem 7.5, which
we shall state without proof.

Theorem7.5:let X 1 , X 2 , .. . , and X n be n normally distributed random variables

(not necessarily independent). Then random variables Y 1 , Y 2 , .. . , and Y m , where
n
X
Y j ˆ c jk X k ; j ˆ 1; 2; ... ; m; 7:40†
kˆ1
are themselves jointly normally distributed.

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