# Poisson distribution

Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article is under development and subject to a disclaimer.

The Poisson distribution is any member of a class of discrete probability distributions named after Simeon Denis Poisson.

It is well suited for modeling various physical phenomena.

## A basic introduction to the concept

### Example

Certain events happen at unpredictable intervals. But for some reason, no matter how recent or long ago last event was, the probability that another event will occur within the next hour is exactly the same (say, 10%). The same holds for any other time interval (say, second). Moreover, the number of events within any given time interval is statistically independent of numbers of events in other intervals that do not overlap the given interval. Also, two events never occur simultaneously.

Then the number of events per day is Poisson distributed.

### Formal definition

Let X be a stochastic variable taking non-negative integer values with probability density function

${\displaystyle P(X=k)=f(k)=e^{-\lambda }{\frac {\lambda ^{k}}{k!}}.}$

Then X follows the Poisson distribution with parameter ${\displaystyle \lambda }$.

### Characteristics of the Poisson distribution

If X is a Poisson distribution stochastic variable with parameter ${\displaystyle \lambda }$, then

• The expected value ${\displaystyle E[X]=\lambda }$
• The variance ${\displaystyle Var[X]=\lambda }$