Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 327

Placing ( 5.148 ) into ( 5.143 ), we get the probability that k events occur in the
interval t ,

k k
k !

k

¼ ð lt Þ e k e lt PfX ¼ k; tg¼ : (5.149)

k !

The probability ( 5.149 ) is another form of the Poisson formula ( 5.143 ) and thus
represents the probability mass function of a Poisson random variable.

Example 5.7.2 Telephone calls can come in to a telephone switching center with
the frequency of 0.5 calls/min. Find the probability that exactly two calls come in,
and the probability that no more than three calls come in.

Solution The number of calls that come in t ¼ 1 min, is a Poisson random variable
with the parameter l ¼ 0.5 calls/min.
From ( 5.149 ), we have:

5 2
2

0 :

e 0 : 5 PfX ¼ 2 ; t ¼ 1 min g¼ ¼ 0 : 0758 : (5.150)

!

; t ¼ 1 min g¼ X

3

PfX 3 P X ðk; tÞ

k¼ 0

5 0
0

5 1
1

5 2
2

0 : 0 : 0 :

e 0 : 5 e 0 : 5 e 0 : 5

¼ þ þ

! ! !

5 3
3

0 :

e 0 : 5 þ ¼ 0 : 9982 : (5.151)

!

5.7.3 Distribution and Density Functions

Using either ( 5.143 )or( 5.149 ), the corresponding distribution and density

functions are:

F X ðxÞ¼ 1
k¼ 0

PfX ¼ kguðx kÞ¼ 1
k¼ 0

k k
k !

e k uðx kÞ

¼ 1
k¼ 0

k

ðltÞ

e lt uðx kÞ;

(5.152)

k !

k k
k !

f X ðxÞ¼ 1
k¼ 0

PfX ¼ kgdðx kÞ¼ 1
k¼ 0

e k dðx kÞ

¼ 1
k¼ 0

k

ðltÞ

e lt dðx kÞ: (5.153)

k !