B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

434 FUNDAMENTALS OF PROBABILITY THEORY
I .
E = 0. Hence, E = 0 and
- 4m p
2 P e (2 2n - 1)
0'2
= E2 = _____ _
E 3 ( 22n )
(8.70b)
Variance of a Sum of Independent Random Variables
The variance of a sum of independent RVs is equal to the sum of their variances. Thus, if x
and y are independent RVs and
then
z=x+y
()'2 = ()'2 + ()' 2
Z X y
(8.71)
This can be shown as follows:
a;: = (z - z) 2 = [x + y - (x + y)] 2
= [(x - x) + (y - y)] 2
= (x - x) 2 + (y - y) 2 + 2(x - x)(y - y)
=a;+ a; + 2(x - x)(y - y)
Because x and y are independent RVs, (x - x) and (y - y) are also independent RVs. Hence,
from Eq. (8.64b) we have
But
(x - x)(y - y) = (x - x) • (y - y)
Similarly,
and
a 2 = a 2 +a 2
Z X y
This result can be extended to any number of variables. If RVs x and y both have zero means
(i.e., x = y = 0), then z = x + y = 0. Also, because the variance equals the mean square value
when the mean is zero, it follows that
z 2 = (x + y) 2 = x 2 + y 2
(8.72)
provided x = y = 0, and provided x and y are independent RVs.