Random Variable - Random Signals and Processes Primer with MATLAB - page 111

Let us summarize the previously mentioned characteristics as:

f X ð 0 Þ¼ 1 ;

(2.389)

j f X ðoÞ j 1 :

Example 2.8.7 We want to find the characteristic function of the exponential
random variable with the density function:

l e lx for x 0 :

f X ðxÞ¼ (2.390)

0 otherwise,

Using ( 2.385 ), the characteristic function is obtained as:

1 1

e jox l e lx d x ¼l 1

l
l jo :

e jox f X ðxÞ d x ¼ e xðljoÞ d x ¼

f X ðoÞ¼ (2.391)

0 0 0

Note that the characteristic function is a complex function. Next, we verify that
the properties in ( 2.389 ) are satisfied:

l
l j 0 ¼ 1

f X ð 0 Þ¼ : (2.392)

l
l 2

f X ðoÞ

j ¼

p 1 :

j (2.393)

þ o 2

Generally speaking, the characteristic functions are complex. However, if the
density function is even, the characteristic function is real.

1

f X ðoÞ¼ cos ðoxÞf X ðxÞ d x; (2.394)

1

From here, the density function is:

1

1
2 p

f X ðxÞ¼ fðoÞ cos ðoxÞ d o: (2.395)

1

Example 2.8.8 Find the characteristic function for the uniform variable in the

interval [ 1, 1].

Solution Using ( 2.385 ), we get:

ð

1

e jox 1

1
2 jo ð e jo

sin ð o Þ
o :

f X ðoÞ¼Ef e joX e jo g¼

2 d x ¼

Þ¼ (2.396)

1