# 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:

1. The number of successes in two disjoint time intervals is independent.

2. 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

• the number of deaths by horse kicking in the Prussian army (first application)

• birth defects and genetic mutations

• rare diseases (like Leukemia, but not AIDS because it is infectious and so not independent) - especially in legal cases

• car accidents

• traffic flow and ideal gap distance

• number of typing errors on a page

• hairs found in McDonald's hamburgers

• spread of an endangered animal in Africa

• 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:

P(X)=(e^{-mu} mu^x)/(x!)

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

1. Some policies

2. 2 or more policies but less than 5 policies.

3. Assuming that there are 5 working days per week, what is the probability that in a given day he will sell one policy?

### 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.

1. Find the probability that none passes in a given minute.

2. What is the expected number passing in two minutes?

3. 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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