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

8.3 Statistical Averages (Means) 429
Figure 8.15
Random
sampling of a
sine-wave
generator.
Sine-wave
generator
p0 (0)
(a)
2-rr
0
2-rr 0 -
(b)
The RV x is thus a function of another RV, 8,
Hence, from Eq. (8.61b),
x = A cos e
and
2n
1
1 1
2n
x= xpe(0)d0 =- Acos0d0 =O
o 2n o
2n
A 2 2n
x 2 = 1 x 2 pe(0) d0 = - 1 cos 2 0 d0
o
2n o
A 2
= 2
Similarly, for the case of two variables x and y, we have
g(x, y) = 1_:1_: g(x, Y)Pxy (x, y)dxdy
(8.62)
Mean of the Sum
lf g1 (x, y), g 2 (x, y), . .. ,g n (X, y) are functions of the RVs x and y, then
g1 (x, y) + g 2 (x, y) + · · · + g n (X, y) = g1 (x, y) + g 2 (x, y) + · · · + g n (X, y)
(8.63a)
The proof is trivial and follows directly from Eq. (8.62).
Thus, the mean ( expected value) of the sum is equal to the sum of the means. An important
special case is
x+y=x + y
(8.63b)
Equation (8.63a) can be extended to functions of any number of RVs.