Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 347

Answer The number of symbols in a group is a geometric random variable
described by the probability ( 5.181 ) where p ¼ 0.5 (see Example 5.8.1). From
( 5.190 ), the mean value of X is:

EfXg¼ 1 =p ¼ 2 : (5.275)

The random variable

P

n
i ¼ 1 X i
n

Y ¼ : (5.276)

From here,

nE f X g
n ¼ EfXg¼ 2

EfYg¼ : (5.277)

Exercise 5.19 In a Poisson flow of calls there is an average number of calls in 1 s,
which is l ¼ 100 calls/s. Find the probability that at least 1 call appears in 1 ms.
Answer The number of calls in t ¼ 10 3 s is described using a Poisson formula
( 5.149 ):

10 3 s g¼ 1 Pfk<

Pfk 1 1 g¼ 1 Pfk ¼ 0 g

;

0

¼ ð 100 10 3

Þ

e 100 10 3

¼ 0 : :

9048 (5.278)

0 !

Exercise 5.20 Find the probability that the time between two successive Poisson
events will be less than its mean value. Also find the mean squared dissipation
around the mean value of the random interval t in which n events have occurred.

Answer The time between two successive Poisson events is an exponential random
variable. Then from ( 5.62 ) and ( 5.65 ), we have:

PfX Xg¼F X ðXÞ¼ 1 e l l ¼ 1 e 1

¼ 0 : 6321 : (5.279)

The random time t is an Erlang's random variable and so, from ( 5.90 ) and
b ¼ n , we have:

n
l 2 :

s 2 ¼ (5.280)