Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 310

Solution Using ( 5.62 ), the desired probability is expressed as:

¼

1
e ¼ 0

1 e P 0 =P 0 PfX > P 0 g¼ 1 PfX P 0 g¼ 1 : 3679 : (5.69)

5.4.3 Memory-Less Property

If past values of a random variable do not affect future values, it is said that the
variable is a memory less .
To this end, consider the probability that the exponential random variable is in the
interval x , if it is known that the variable has been previously in the interval x 0 ,asshown
in Fig. 5.10 . This statement can be expressed in a mathematical form as in ( 5.70 ).

P f x 0 < X x 0 þ x 1 g
PfX > x 0 g

F X ð x 0 þ x 1 Þ F X ð x 0 Þ
1 F X ðx 0 Þ

PfX x 0 þ x 1 j X > x 0 g¼ ¼ : (5.70)

Using ( 5.62 ), we arrive at:

j X > x 0 g¼ ð 1 e lðx 0 þx 1 Þ Þð 1 e lx 0

Þ

¼ 1 e lx 1 PfX x 0 þ x 1 : (5.71)

1 ð 1 e lx 0

Þ

The obtained result ( 5.71 ) indicates that the desired probability does not depend
on the length of x 0 , but on the length of x 1 :

j X > x 0 g¼PfX x 1 g¼ 1 e lx 1 PfX x 0 þ x 1 : (5.72)

5.5 Variables Related with Exponential Variable

5.5.1 Laplacian Random Variable

A random variable is said to be a Laplacian random variable if its density function
is defined as:

f X ðxÞ¼k e ljxj ; for 1<x<1; (5.73)

where k is the constant and l is the positive parameter.

Fig. 5.10 Memory-less
property of the exponential
variable