Download Poisson

MATH 1107
Introduction to Statistics
Lecture 10
The Poisson Distribution
Math 1107 Poisson Distribution
What if we are interested in obtaining the
probability of a “success” when the number of
“failures” is potentially infinite? Such as the
probability of a web site hit? Or the probability
of being in a car accident?
Math 1107 Poisson Distribution
A poisson probability distribution results from a
procedure that meets all the following requirements:
 The random variable x is the number of occurrences
of an event over some interval.
 The occurrences must be random.
 The occurrences must be independent of each other.
 The occurrences must be uniformly distributed over
the interval being used.
Math 1107 Poisson Distribution
P(x) =
µ •e
x
-µ
x!
where e  2.71828
µ = average number of occurrences
X is the occurrence of interest
Math 1107 Poisson Distribution
The Poisson distribution differs from the binomial
distribution in these fundamental ways:
• The binomial distribution is affected by the sample size
n and the probability p, whereas the Poisson
distribution is affected only by the mean μ.
• In a binomial distribution the possible values of the
random variable are x are 0, 1, . . . n, but a Poisson
distribution has possible x values of 0, 1, . . . , with no
upper limit.
Math 1107 Poisson Distribution
World War II Bombs In analyzing hits by V-1 buzz bombs in
World War II, South London was subdivided into 576 regions,
each with an area of 0.25 km2. A total of 535 bombs hit the
combined area of 576 regions
If a region is randomly selected, find the probability that it was
hit exactly twice.
The Poisson distribution applies because we are dealing with
occurrences of an event (bomb hits) over some interval (a
region with area of 0.25 km2).
Math 1107 Poisson Distribution
The mean number of hits per region is
μ
P(x) =
number of hits
number of regions
535

 0.929
576
.9292 • e -929
2!
The probability of a particular region being hit exactly
twice is P(2) = 0.170.
Fun EXCEL Exercise
Math 1107 Poisson Distribution
Example 2 –
For a period of 100 years, there were 93 major earthquakes in
the world. What is the probability that the number of
earthquakes in a randomly selected year is 5?
P(x) =
.93 5 • e -.93
5!
=
.00229
Math 1107 Poisson Distribution
Example 3 –
A certain machine process generates 1 defect for every 200
units produced per day. What is the probability of generating
exactly 3 defects in a single day?
P(x) =
.0023 • e -.002
3!
=
.00000000207
Math 1107 Poisson Distribution
Example 4 –
You work for a large property insurance company in Florida.
You need to determine the needed cash reserves for the
upcoming hurricane season. You know that in the last 52 years,
Florida has been hit with 72 hurricanes. Each hurricane
generates approximately $10M in claims. What is the probability
that this year, Florida will experience 3 hurricanes?
P(x) =
1.3853 • e –1.385
3!
=
.11084