Lecture 7: The Poisson Distribution
IE 111 Fall Semester 2011
The Poisson Distribution
The Poisson Distribution is another extremely important distribution in Probability and
Statistics.
It is especially important for Industrial Engineers for reasons that will
become apparent.
The distribution was discovered by SiméonDenis Poisson
(1781
–1840
) and published,
together with his probability theory, in 1838
in his work
Recherches sur la probabilité
des jugements en matières criminelles et matière civile
("Research on the Probability of
Judgments in Criminal and Civil Matters").
The Poisson distribution describes the probability that a random event will occur in a
time or space interval under the conditions that the probability of the event occurring is
very small, but the number of trials is very large so that the event actually occurs a few
times.
The word “Poisson” is a French word, thus is pronounced differently depending on the
ability to speak French.
Common pronunciations include:
1. Poy saan
2. Pwah saan
3. Pwah son
Since 1. is really a butchery of French and 3. requires a French accent thereby appearing
snooty, I usually use 2.
The Poisson distribution is derived from the Binomial by letting N
→∞
and p
→
0 but all
the time keeping Np=
λ
constant.
For example:
N
10
100
1000
10,000

.......
λ
= 1 always
p
0.1
0.01
0.0001
0.00001
In the limit, the Binomial approaches the Poisson (see book for details).
The Poisson
PMF is given by:
P
X
(x) =
e

λ
λ
x
/ x!
for x in {0,1,2,.
....}
Note that the domain includes all nonnegative integers, and that the Poisson
distribution has a single parameter
λ
.
The mean and variance of the Poisson can be shown (after a lot of tedious math) to be:
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View Full DocumentE(X) =
λ
and V(X) =
λ
λ
=
=
=

=

=
=
=
∑
∑
∑
∑
∑
∞
=

∞
=


∞
=

∞
=

∞
=

)
1
(
!
)
(
!
)
1
(
!
)
1
(
!
!
)
(
0
1
1
1
1
0
k
k
j
j
j
j
j
j
j
j
k
e
j
e
j
e
j
je
j
je
X
E
Derivation of the variance is similar, and is in the book.
The fact that the mean and variance are equal is an unusual, but not terribly useful
result.
One use of the Poisson distribution is as an approximation to the Binomial.
As we have
seen, one difficulty of using the Binomial is that it is quite time consuming to calculate
the CDF in order to evaluate P(X
≤
x).
Example
Suppose 100 chips are manufactured on a line which has historically produced 5%
defective chips.
What is the probability that 5 or fewer of the 100 are defective?
Letting X = the number of defective chips, it should be clear by now that X is a
Binomial random variable with N=100 and p=0.05.
We want to find P(X
≤
5).
Using the Binomial,
P(X
≤
5) = P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)
=
100
C
0
0.05
0
(0.95)
100
+
100
C
1
0.05
1
(0.95)
99
+
100
C
2
0.05
2
(0.95)
98
+
100
C
3
0.05
3
(0.95)
97
+
100
C
4
0.05
4
(0.95)
96
+
100
C
5
0.05
5
(0.95)
95
=
0.0059205 + 0.0311606 + 0.0811817 + 0.1395756 + 0.1781426 + 0.1800178
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 Fall '07
 Storer
 Poisson Distribution, Probability theory, Binomial distribution, Poisson

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