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

Continues below

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?