Poisson Distribution

The Poisson distribution is used to model the number of occurrences of a relatively rare event in a fixed interval of time, length, area or volume — for example, the number of customers arriving at a bank counter in a minute, the number of accidents at a road junction in a week, the number of printing errors per page of a book, or the number of defective items in a large batch when the defect rate is small.

The Poisson distribution can be obtained as a limiting form of the Binomial distribution when the number of trials $n$ is very large, the probability of success $p$ is very small, but the product $np = \lambda$ (a finite constant, called the mean number of occurrences) stays fixed. In such situations it is far easier to use the Poisson formula than to compute binomial probabilities with a very large $n$.

A discrete random variable $X$ follows a Poisson distribution with parameter (mean) $\lambda > 0$ if its pmf is:

where $e \approx 2.71828$ is the base of natural logarithms. Values of $e^{-\lambda}$ for common $\lambda$ are usually available from a table or calculator and are generally given in board-exam numericals so that arithmetic stays manageable.

Mean and variance of the Poisson distribution. A distinguishing feature of the Poisson distribution is that its mean and variance are equal:

$E(X) = \lambda, \qquad Var(X) = \lambda$ …