# Poisson Distribution

Poisson distribution was introduced by a French mathematician Simon Denis Poisson in 1837.

It is a limiting form of binomial distribution. Poisson distribution is applied in situations where the probability of success is very low and that of failure is very high in a small interval of time.

For example, a TV manufacturer knows that on an average 5% of the televisions manufactured by him are defective. Here the probability of success, *i.e.*, the probability to find a defective TV set is very low. Hence, we can apply Poisson distribution in this case to find the probability that a TV set is not defective.

# Conditions Under which Poisson Distribution can be Applied

- The number of trials conducted must be large.
- The probability of success must be very small.
- The trials must be independent.
- The variables are discrete.
- Mean of the distribution is finite and moderate.

# Properties of Poisson Distribution

Let *X* be a Poisson variate with parameter *m*. Then,

- The mean of the Poisson distribution is
*m*=*np*. - Variance of the Poisson distribution = Mean =
*m*. - Standard deviation S.D. =
- If
*x*and*y*are two independent random variables following Poisson distribution with parameters*m*_{1}and*m*_{2}, then*x*+*y*also follows Poisson distribution with a parameter*m*_{1 }+*m*_{2}.*x*~*P*(*m*_{1}) and*y*~*P*(*m*_{2}), then (*x*+*y*) ~*P*(*m*_{1 }+*m*_{2}). - A random variable
*X*is said to follow Poisson distribution if its probability mass function is given by where m is the mean of the Poisson.

**Note:**

- If a random variable x follows Poisson distribution with the parameter m, it is represented as x ~ P(m).
- Poisson distribution is also known as uni-parametric distribution as it depends on only 1 parameter m.

# Poisson Distribution Approximation to Binomial Distribution

Consider a binomial distribution B(n, p). If in this distribution, the number of trials n tends to infinity and the probability of success p tends to 0, such that the mean m = np is kept finite, then this binomial distribution can be approximated to a Poisson distribution with the parameter m.

So, B(n, p) = P(m).

# Application of Poisson Distribution

It is applied where the total number of events is large, but the probability of occurrence is very small.

For example:

- Number of print mistakes per page in a text book
- Number of defective bulbs manufactured by a reputed firm
- Number of accidents on a road during busy hours
- Number of suicides reported in a year in a particular city

Example

What probability model is appropriate to describe a situation where 100 misprints are distributed randomly throughout the 100 pages of the book? For this model, what is the probability that a page observed at random will contain at least 3 misprints?

Solution

Since 100 misprints are distributed randomly throughout the 100 pages of a book, on an average there is only one mistake in a page. This means, the probability of their being a misprint,
Average number of misprints in one page

*ρ*= 1/100, is very small and the number of words,*n*in 100 pages are very large. Hence, Poisson distribution is best suited in this case.

Example

Suppose a life insurance company insures the lives of 500 persons aged 42. If studies show that the probability of any 42-year-old person will die in a given year to be 0.001, find the probability that the company will have to pay at least two claims during a given year.

Solution

Given that

*n*= 5000,*p*= 0.001, so . Thus, the probability the company will have to pay at least 2 claims during a given year is

Example

In a town, 10 accidents took place in the span of 50 days. Assuming that the number of accidents per day follows the Poisson distribution, find the probability that there will be 3 or more accidents in a day.

Solution

The average no. of accidents per day = 10/50 = 0.2.