Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 340

Exercise 5.6
(a) Find the probability that a random time between two consecutive Poisson
events with a constant intensity of l will be less than its mean value.
(b) Find the mean squared deviation from its mean value for the time interval t in
which k Poisson events occur.
Answer
(a) The time between two consecutive Poisson events is an exponential random
variable X . From ( 5.65 ),

EfXg¼ 1 =l; (5.224)

=lÞ¼ 1 e 1 PfX< 1 =lg¼F X ð 1 ¼ 0 : 6321 : (5.225)

(b) A mean squared deviation from a mean value is a variance of random variable.
The time interval in which k Poisson events occur is an Erlang's variable.
From ( 5.90 ), the variance of a Gamma variable is

b
l 2 :

s 2

¼

(5.226)

Replacing b with k in ( 5.226 ), we get the variance of Erlang's variable:

k
l 2 :

s 2 ¼ (5.227)

Exercise 5.7 A random variable X has the PDF

f X ðxÞ¼ e x uðxÞ:

(5.228)

Find the PDF and mean value of the random variable Y if

aX for X 0
0

Y ¼ (5.229)

otherwise a > 0 :

;

Answer The PDF of the random variable Y is obtained from (2.138), as:

y
a ;

f X ð x Þ
a

f X ð y = a Þ
a

1
a

e f Y ðyÞ¼ ¼ ¼ for y 0 : (5.230)

f Y ðyÞ¼ 0 for y< 0 ; (5.231)

The variable X is an exponential variable for which l ¼ 1. As a consequence,

EfXg¼ 1 =l ¼ 1 : (5.232)

The mean value of the variable Y is:

EfYg¼EfaXg¼aEfXg¼a:

(5.233)