Page 102 - Fundamentals of Communications Systems

3.16 Chapter Three

A majority of the results presented in this section correspond to two random
variables. The extension of these concepts to three or more random variables
is straightforward.

3.3.2 Joint Moments and Statistical Averages
Joint moments and statistical averages are also of interest in communication
system engineering. The general statistical average of a function g(X, Y ) of two
random variables X and Y is given as

∞ ∞
E[g(X, Y )] = g(x, y) p XY (x, y)dxdy
−∞ −∞
A commonly used joint moment or statistical average is the correlation be-
tween two random variables X and Y, defined as
∞ ∞

E[XY ] = corr(X, Y ) = xyp XY (x, y)dxdy
−∞ −∞
A frequently used joint central moment is the covariance between two random
variables x and y, defined as

∞ ∞
E[(X − m X )(Y − m Y )] = cov(X, Y ) = (x − m X )(y − m Y ) p XY (x, y)dxdy
−∞ −∞
Definition 3.14 The correlation coefficient is
cov(X, Y ) E[(X − m X )(Y − m Y )]
ρ XY = √ =
var(X)var(Y ) σ X σ Y
The correlation coefficient is a measure of the statistical similarity of two
random variables. If |ρ XY |= 1 for random variables X and Y , then X is a
scalar multiple of Y .If ρ XY = 0 then the random variables are uncorrelated.
Values of |ρ XY | between these two extremes provide a measure of the similarity
of the two random variables (larger |ρ XY | being more similar).

3.3.3 Two Gaussian Random Variables
Two jointly Gaussian random variables, X and Y , have a joint density
function of
1
f XY (x, y) =
2πσ X σ Y 1 − ρ 2 XY
2 2
1 (x − m X ) 2ρ XY (x − m X )(y − m Y ) (y − m Y )
× exp − 2 2 − + 2
2(1 − ρ XY ) σ X σ X σ Y σ Y
where m X = E(X), m Y = E(Y ), σ X 2 = var(X), σ Y 2 = var(Y ), and ρ XY is the
correlation coefficient between the two random variables X and Y . This density
function results in a three-dimensional bell-shaped curve which is stretched and