# 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

1. The number of trials conducted must be large.
2. The probability of success must be very small.
3. The trials must be independent.
4. The variables are discrete.
5. 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 m1 and m2, then x + y also follows Poisson distribution with a parameter m1 m2.

So, if x ~ P(m1) and y ~ P(m2), then (x y) ~ P(m1 m2).
• 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, 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.

Average number of misprints in one page

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.