The Poisson distribution named after French mathematician Siméon Denis Poisson (1781-1840).

is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event

Examples of the Poisson distribution are: the number of cars passing in a certain place during one hour, number of phone calls etc.

Probability of events for a Poisson distribution

An event can occur 0, 1, 2, … times in an interval. The average number of events in an interval is designated $\lambda$ (lambda). Lambda is the event rate, also called the rate parameter. The probability of observing k events in an interval is given by the equation

- $P(k{\text{ events in interval}})=e^{-\lambda }{\frac {\lambda ^{k}}{k!}}$

where

$\lambda$ is the average number of events per interval

*e* is the number 2.71828... (Euler's number) the base of the natural logarithms

*k* takes values 0, 1, 2, …

*k*! = *k* × (*k* − 1) × (*k* − 2) × … × 2 × 1 is the factorial of *k*.

For example: Salesman sales, in average, 5 products per week. The chance he will sale, in a given week 10 products is 1.8133%

(λ=2; k=10)