The Poisson Distribution

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MATH 1107Introduction to Statistics

• Lecture 10
• The Poisson Distribution

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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?

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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.

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Math 1107 Poisson Distribution
where e ? 2.71828 µ average number of
occurrences X is the occurrence of interest
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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.

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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).
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Math 1107 Poisson Distribution
The probability of a particular region being hit
exactly twice is P(2) 0.170.
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Fun EXCEL Exercise
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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?
.00229

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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?
.00000000207

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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?
.11084