13. The Poisson Probability Distribution
The Poisson Distribution was developed by the French mathematician Simeon Denis Poisson in 1837.
The Poisson random variable satisfies the following conditions:
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The number of successes in two disjoint time intervals is independent.
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The probability of a success during a small time interval is proportional to the entire length of the time interval.
Apart from disjoint time intervals, the Poisson random variable also applies to disjoint regions of space.
Applications
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the number of deaths by horse kicking in the Prussian army (first application)
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birth defects and genetic mutations
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rare diseases (like Leukemia, but not AIDS because it is infectious and so not independent) - especially in legal cases
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car accidents
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traffic flow and ideal gap distance
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number of typing errors on a page
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hairs found in McDonald's hamburgers
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spread of an endangered animal in Africa
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failure of a machine in one month
The probability distribution of a Poisson random variable X representing the number of successes occurring in a given time interval or a specified region of space is given by the formula:
where
x = 0, 1, 2, 3...
e = 2.71828 (but use your calculator's e button)
μ = mean number of successes in the given time interval or region of space
Mean and Variance of Poisson Distribution
If μ is the average number of successes occurring in a given time interval or region in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to μ.
E(X) = μ
and
V(X) = σ2 = μ
Note: In a Poisson distribution, only one parameter, μ is needed to determine the probability of an event.
EXAMPLE 1
A life insurance salesman sells on the average 3 life insurance policies per week. Use Poisson's law to calculate the probability that in a given week he will sell
(a) some policies
(b) 2 or more policies but less than 5 policies.
(c) Assuming that there are 5 working days per week, what is the probability that in a given day he will sell one policy?
Here is a neat Java applet which can find these values for you:
Poisson applet. (External site).
EXAMPLE 2
Twenty sheets of aluminum alloy were examined for surface flaws. The frequency of the number of sheets with a given number of flaws per sheet was as follows:
| Number of flaws | Frequency |
|---|---|
| 0 | 4 |
| 1 | 3 |
| 2 | 5 |
| 3 | 2 |
| 4 | 4 |
| 5 | 1 |
| 6 | 1 |
What is the probability of finding a sheet chosen at random which contains 3 or more surface flaws?
EXAMPLE 3
If electricity power failures occur according to a Poisson distribution with an average of 3 failures every twenty weeks, calculate the probability that there will not be more than one failure during a particular week.
EXAMPLE 4
Vehicles pass through a junction on a busy road at an average rate of 300 per hour.
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Find the probability that none passes in a given minute.
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What is the expected number passing in two minutes?
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Find the probability that this expected number actually pass through in a given two-minute period.
EXAMPLE 5
A company makes electric motors. The probability an electric motor is defective is 0.01. What is the probability that a sample of 300 electric motors will contain exactly 5 defective motors?
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