Fundamentals of Probability and Statistics for Engineers

Let us verify Result 4.3 for n 2. The proof for the case of n random
variables follows at once by mathematical induction. Consider

We know from Equation (4.38) that

Subtracting mY from Y, and (m 1 m 2 ) from (X 1 X 2 ) yields

and

The covariance cov(X 1 ,X 2 ) vanishes, since X 1 and X 2 are independent [see
Equation (4.27)], thus the desired result is obtained.
Again, many generalizations are possible. For example, if Z is given by
Equation (4.39), we have, following the second of Equations (4.9),

Let us again emphasize that, whereas Equation (4.38) is valid for any set of
random variables X 1 ,...Xn, Equation (4.41), pertaining to the variance, holds
only under the independence assumption. Removal of the condition of inde-
pendence would, as seen from the proof, add covariance terms to the right-
hand side of Equation (4.41). It would then have the form

Expectations and Moments 95

^2 Yˆ^21 ‡^22 ‡‡^2 n: … 4 : 41 †

ˆ

YˆX 1 ‡X 2 :

mYˆm 1 ‡m 2 :

‡ ‡

YmYˆ…X 1 m 1 †‡…X 2 m 2 †

^2 YˆEf…YmY†^2 gˆEf‰…X 1 m 1 †‡…X 2 m 2 †Š^2 g

ˆEf…X 1 m 1 †^2 ‡ 2 …X 1 m 1 †…X 2 m 2 †‡…X 2 m 2 †^2 g

ˆEf…X 1 m 1 †^2 g‡ 2 Ef…X 1 m 1 †…X 2 m 2 †g‡Ef…X 2 m 2 †^2 g

ˆ^21 ‡2co v…X 1 ;X 2 †‡^22 :

^2 Zˆa^21 ^21 ‡‡a^2 n^2 n: … 4 : 42 †

^2 Yˆ^21 ‡^22 ‡‡^2 n‡2 cov…X 1 ;X 2 †‡2 cov…X 1 ;X 3 †‡‡2 cov…Xn 1 ;Xn†

ˆ

Xn

jˆ 1

^2 j‡ 2

Xn^1

iˆ 1

i<j

Xn

jˆ 2

cov…Xi;Xj†… 4 : 43 †