Page 96 - Fundamentals of Communications Systems

3.10 Chapter Three

The most commonly used second central moment is the variance, σ X,2 , which
provides a measure of the spread of a random variable around the mean. The
2
variance has the shorthand notation σ X,2 = σ = var(X). The relation between
X
the variance and the mean square value is given by
2
2
E(X ) = m + σ 2
X X
The expectation operator is linear, i.e.,
E(g 1 (x) + g 2 (x)) = E(g 1 (x)) + E(g 2 (x))

EXAMPLE 3.10
Consider again the rand(•) function in Matlab as an example of a random variable.
Since
⎧
1
a ≤ x ≤ b
⎨
f X (x) = b − a (3.18)
0 elsewhere
⎩
the mean is given as
b 2 2

1 b − a b + a
E[X] = xdx = = (3.19)
b − a 2(b − a) 2
a
and the variance is
(b − a) 2
var(X) = (3.20)
12
3.2.4 The Gaussian Random Variable
The most common RV encountered in communications system analysis is the
Gaussian random variable. This RV is a very good approximation to many
of the physical phenomena that occur in electronic communications (e.g., the
noise voltage generated by thermal motion of electrons in conductors). The
reason for the common usage of the Gaussian RV is the Central Limit Theorem
(see Section 3.3.5) and that analysis with Gaussian RVs is often tractable. This
section will review the characteristics of the Gaussian RV. A Gaussian or normal
2
2
RV, X, of mean m X and variance σ is denoted N (m X , σ ). The Gaussian RV is a
X X
continuous random variable taking values over the whole real line and a PDF of
1 (x − m X ) 2
exp −
f X (x) = 2
2πσ X 2 2σ X
The Gaussian RV is characterized completely by its mean and variance. The
Gaussian PDF is often referred to as the bell-shaped curve. The mean shifts the
centroid of the bell-shaped curve as shown in Figure 3.3(a). The variance is a
measure of the spread in the values the random variable realizes. A large vari-
ance implies that the random variable is likely to take values far from the mean,