2.12. SOME TYPICAL PROBABILITY DISTRIBUTIONS 67
2.12.2 POISSON DISTRIBUTION
In many engineering problems, the occurrence of an event is only affected by chance. Some
examples are: (a) fatigue cracks that may occur in an automobile transmission shaft; (b) the
number of telephone calls that may be received at any time over a specified period of time; and
(c) the number of automobiles that may arrive at a tollbooth at any time over a specified period
of time. e occurrence of such an event can be described by Poisson distribution.
Poisson Distribution: If an event occurs randomly and independently at any time or any
point in space with the same likelihood at any subinterval, the random variable X , that denotes
the number of events in an interval, can be described by the Poisson distribution. e PMF of
Poisson distribution is
p
.
X D x
/
D
e
x
xŠ
I x D 0; 1; 2; : : : ; (2.62)
where X is the number of occurrence events in the interval and x is the realizing value of the
discrete random variable X. is the mean rate of occurrence of the event per the specified
interval.
e Poisson distribution is fully specified by one distribution parameter . ere are some
notes about . For example, if the number of defects of a manufacturing bar per inch is 0.0001,
the for a 20
00
-long bar will be:
D 0:0001 20 D 0:002 .defects per the 20
00
bar/:
In this example, the event is an occurrence of defects in a 20
00
-long bar, and the interval is not
specified and is “present,” that is, the present after the bars has been manufactured. e for a
5
00
-long bar will be
D 0:0001 5 D 0:0005 .defects per the 5
00
bar/:
For another example, if the average number of defects in a long underground cable per 1000-m
length per year is 0.07, the for a 500-m length in a four year will be
D
0:07
1000 1
.
500 4
/
D 0:14 .defects per the 500-m length per 4 years/:
In this example, the event is defects of the 500-m length cable in 4 years.
e CDF of a Poisson distribution per Equation (
2.43) will be:
F
.
x
/
D P
.
X x
/
D
kDx
X
kD1
e
k
kŠ
I x D 0; 1; 2; : : : : (2.63)
e mean and standard deviation of the Poisson distribution with a distribution parameter is
X
D E
.
X
/
D (2.64)