Random Variable - Random Signals and Processes Primer with MATLAB - page 88

we arrive at:

# "

5 1
0

1
0 ¼

e 0 : 5 x
ð 0

0 : 5

x e 0 : 5 x d x ¼ 0 m ¼ 0 : : 5

2 ð 0

: 5 x 1 Þ

5 2 ¼ 2

: (2.269)

:

: 5 Þ 0

The following condition must be satisfied in order for a mean value to exist (the
integral at the right side of ( 2.261 ) must converge absolutely)

1

j xj f X ðxÞ d x <1: (2.270)

1

If the condition in ( 2.270 ) is not satisfied, the mean value of the random variable
does not exist, as shown in the next example.

Example 2.7.10 Let X be a continuous random variable in the range 1< x <1
with the density function:

1
pð 1 þ x 2

f X ðxÞ¼

Þ ;

1< x <1: (2.271)

A random variable that has this density function is called a Cauchy random
variable . However, in Cauchy random variables, the condition in ( 2.270 ) is not
satisfied (see Integral 15, Appendix B ) :

1 1

1
p

jxj
1 þ x 2 d x ¼

2
p

x
1 þ x 2 d x ¼ lim

1
p ln ð 1 þ x 2

Þ¼1:

(2.272)

X!1

1 0

Consequently, for the Cauchy random variable, the mean value E { X } does not
exist.
Similarly, as in the case of a discrete random variable, we may consider the
function g ( X ) instead of the random variable X itself. From ( 2.270 ), we obtain:

1

EfgðXÞg ¼ gðxÞf x ðxÞ d x: (2.273)

1

Example 2.7.11 Consider the expected value of 2 X 2 + 1 for the random variable X
with the density function:

(

1 = 3 for 1 x 2 ;

f X ðxÞ¼

(2.274)

0 otherwise,

as shown in Fig. 2.44 .