Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 313

The mean and mean squared values are obtained from the characteristic function
( 5.81 ) using the moment theorem:

o¼ 0 ¼ 0

1
j

d f X ð o Þ
d o

X ¼ :

(5.82)

o¼ 0 ¼

d 2 f X ð o Þ
d o 2

1
j 2

2
l 2 :

X 2 ¼ (5.83)

From here the variance is:

2
l 2 :

X 2 s 2

¼ X 2 ¼ (5.84)

The Laplacian randomvariable is useful inmodeling a speech signal [MIL04, p. 66].

5.5.2 Gamma and Erlang's Random Variables

5.5.2.1 Gamma Variable

As opposed to an exponential random variable with only one parameter, sometimes
it is easier to use variables related with the exponential random variable but which
have more parameters. One such variable is a Gamma variable .
Its density function is defined as:

8
<

l b
GðbÞ

x b 1 e lx

x 0 ;

for

f X ðxÞ¼ (5.85)

:

0 otherwise ;

where l and b are positive parameters and GðbÞ is the Gamma function , defined as:

1

y b 1 e y d y: GðbÞ¼ (5.86)

0

The properties of Gamma function are given in Appendix C .
The Gamma densities for l ¼ 1, and the different values of the parameter b are
shown in Fig. 5.12 . Note that the Gamma variable becomes the exponential variable
for b ¼ 1.