Fundamentals of Probability and Statistics for Engineers

Theorem 7. 4: let X 1 ,X 2 ,...,Xn be n jointly normally distributed random
variables (not necessarily independent). Then random variable Y, where

is normally distributed, where andare constants.

Proof of Theorem 7.4:for convenience, the proof will be given by assuming
that all Xj,j 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),

where ij cov(Xi,Xj).
Since Xj are normally distributed, their jo int characteristic function is given
by Equation (7.33), which is

The characteristic function of Y is

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.

The orem 7. 5 : let X 1 ,X 2 ,..., andXn be n normally distributed random variables
(not necessarily independent). Then random variables Y 1 ,Y 2 , ..., andYm,where

are themselves jointly normally distributed.

208 Fundamentals of Probability and Statistics for Engineers

Yˆc 1 X 1 ‡c 2 X 2 ‡‡cnXn; … 7 : 36 †

c 1 ,c 2 ,, cn

ˆ

^2 YˆEfY^2 gˆ

Xn

iˆ 1

Xn

jˆ 1

cicjij; … 7 : 37 †

 ˆ

X…t†ˆexp 

1

2

Xn

iˆ 1

Xn

jˆ 1

ijtitj

!

: … 7 : 38 †

Y…t†ˆEfexp…jtYgˆE exp jt

Xn

kˆ 1

ckXk

)!

ˆexp 

1

2

t^2

Xn

iˆ 1

Xn

jˆ 1

ijcicj

!

ˆexp…

1

2

^2 Yt^2 †; … 7 : 39 †

Yjˆ

Xn

kˆ 1

cjkXk; jˆ 1 ; 2 ;...;m; … 7 : 40 †