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Poisson Distribution: Understanding Probability of Occurrences with Examples, Study notes of Probability and Statistics

The concept of the poisson distribution, a probabilistic model used to describe the number of occurrences in a given time or space interval. With examples, it covers the relationship between the mean number of occurrences and the probability of a specific number of occurrences in a minute, and the connection to the exponential distribution.

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

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Econ 209
Fall 2002
Handout # 2
The Poisson Distribution
The Poisson Distribution: Suppose that the probability of an occurrence is the same for any
equally sized interval of time or space and is independent of the number of occurrences in any
other interval of time or space. Let be the mean number of occurrences in a given time or
space interval. Let x be the number of occurrences. The it can be shown that x has a Poisson
distribution:
Example I: Suppose that there are an average of 600 hits per hour on a popular web site. What
is the probability that in a given minute 20 hits occur. The average number of hits per minute
is equal to 10.
Example II: Suppose a switchboard can handle up to 2 calls per minute. Suppose that the
switchboard receives 30 calls per hour on the average. What is the probability that in any given
minute the switchboard is overloaded. a call per minute.
Example III: A disease occurs on the average in 1 person in 100,000. What is the probability that
there are 10 cases of the disease in a city of 500000. .
Note: An interesting feature of the Poisson distribution is that
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Econ 209 Fall 2002 Handout # 2

The Poisson Distribution

The Poisson Distribution : Suppose that the probability of an occurrence is the same for any equally sized interval of time or space and is independent of the number of occurrences in any

other interval of time or space. Let be the mean number of occurrences in a given time or space interval. Let x be the number of occurrences. The it can be shown that x has a Poisson distribution:

Example I: Suppose that there are an average of 600 hits per hour on a popular web site. What

is the probability that in a given minute 20 hits occur. The average number of hits per minute is equal to 10.

Example II: Suppose a switchboard can handle up to 2 calls per minute. Suppose that the switchboard receives 30 calls per hour on the average. What is the probability that in any given

minute the switchboard is overloaded. a call per minute.

Example III: A disease occurs on the average in 1 person in 100,000. What is the probability that

there are 10 cases of the disease in a city of 500000..

Note: An interesting feature of the Poisson distribution is that

The exponential distribution is closely related to the Poisson distribution. Suppose that x is has a Poisson distribution. Let t = the time interval between occurrences. Then the probability density function of t is given by

The cumulative density function is

The cumulative density function of a random variables give the probability that the random variable is less than or equal to a particular value. For example

Example: Suppose that the average number of cars passing a given spot is 30 car per hour. What is the probability that there is a five minute time interval with no cars.

P(t > 5) = 1 - F(5) =