Random Variable - Random Signals and Processes Primer with MATLAB - page 115

From here, we can find moments from the corresponding characteristic
functions:

o¼ 0

1
j

d f ð o Þ
d o

m 1 ¼

...

o¼ 0

d n f ð o Þ
d o n

1
j n

m n ¼ (2.415)

The expressions ( 2.415 ) are also known as the moment theorem.

Example 2.8.11 Find the mean value and variance of the exponential variable from
Example 2.8.7, using the moment theorem.

Solution Based on the result ( 2.391 ), the characteristic function is:

l
l jo :

f X ðoÞ¼ (2.416)

From ( 2.415 ) and ( 2.416 ), we have:

o¼ 0 ¼

1
j

d f ð o Þ
d o

1
l

m 1 ¼

o¼ 0 ¼

d 2 f ð o Þ
d o 2

1
j 2

2
l 2 :

m 2 ¼ (2.417)

This is the same result as ( 2.377 ) and ( 2.378 ).
The variance is:

2
l 2

1
l 2 ¼

1
l 2 ;

s 2 ¼ m 2 m 1 ¼ (2.418)

which is the same result as in ( 2.379 ).
The corresponding equation for the characteristic function of the discrete
random variable is:

f X ðoÞ¼ X
i

e jox i PðX ¼ x i Þ: (2.419)

The use of ( 2.419 ) is illustrated in the following example.

Example 2.8.12 Find the characteristic function of a discrete variable X which
takes with the equal probabilities each of its possible values: C and C .

PfX ¼ Cg¼PfX ¼Cg¼ 1 = 2 : (2.420)