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

430 FUNDAMENTALS OF PROBABILITY THEORY
Mean of the Product of Two Functions
Unfortunately, there is no simple result [as in Eq. (8.63)] for the product of two functions. For
the special case where
g(x, y) = g1 (x)g 2 (y)
(8.64a)
g1 (x)g 2 (y) = 1_:1_: g1 (x)g2 (Y)Pxy (x, y) dxdy
If x and y are independent, then [Eq. (8.53c)]
and
Pxy (x, y) = Px(x)py (y)
if x and y independent
(8.64b)
A special case of this is
xy = xy if x and y independent (8.64c)
Moments
The nth moment of an RV x is defined as the mean value of x n . Thus, the nth moment of x is
(8.65a)
The nth central moment of an RV x is defined as
(8.65b)
The second central moment of an RV x is of special importance. It is called the variance of
x and is denoted by a;, where a x is known as the standard deviation (SD) of the RV x. By
definition,
a; = (x - x) 2
= x 2 - 2xx + x 2 = x 2 - 2x 2 + x 2
= x 2 _ x: 2
(8.66)
Thus, the variance of x is equal to the mean square value minus the square of the mean. When
the mean is zero, the variance is the mean square; that is, x 2 = a; .