Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 311

Applying the condition that the area below a PDF is equal to the unity,

1 ð 1

0

2 k
l ¼ 1

k e lx d x þ k e lx d x ¼ f X ðxÞ d x ¼ ; (5.74)

1 1 0

the constant k is related with the parameter l as:

k ¼ l= 2 : (5.75)

Using ( 5.75 ), the density function ( 5.73 ) becomes

l
2 e ljxj

f X ðxÞ¼ ; for 1<x<1: (5.76)

The variable for l ¼ 0.5 is shown in Fig. 5.11a . Note that the density shown in
Fig. 5.11b takes the form of a two-sided exponential PDF with a peak which is one-
half of that of the exponential PDF.
The corresponding distribution function is

ð

x

l
2 e ljxj d x:

F X ðxÞ¼ (5.77)

1

For x 0, we have:

ð

x

l
2

1
2 e lx

e lx d x ¼ F X ðxÞ¼ : (5.78)

1

For x 0, it follows:

2
4

3
5 ¼ 1

ð ð

0 x

l
2

1
2 e lx

e lx d x þ e lx d x

F X ðxÞ¼ :

(5.79)

1 0

From ( 5.78 ) and ( 5.79 ), we obtain:

5e lx 0 : for x 0 ;

F X ðxÞ¼ (5.80)

5e lx

1 0 : for x 0 :