Poisson distribution

A discrete probability distribution for counts of independent events in a fixed interval, determined by a single rate parameter λ; used to model rare or random event counts.

Overview

The Poisson distribution is a discrete probability model commonly used in statistics to describe the number of times an event occurs in a fixed interval of time or space when events occur independently and with a constant average rate. It applies to counts that can take the values 0, 1, 2, … and is appropriate when events are relatively rare compared with the size of the observation window. $\lambda$

Definition and formula

When a nonnegative integer random variable X follows a Poisson distribution with parameter λ (lambda), the probability that X equals x is given by the probability mass function (PMF): P(X = x) = e^-λ λ^x / x! for x = 0, 1, 2, … . The parameter λ > 0 represents both the expected value and the variance of the distribution. This simple form makes the Poisson especially convenient for modeling and inference in many applied settings. $X$

Key properties

Mean and variance: Both equal λ, so dispersion is tied directly to the rate.
Support: Nonnegative integers only, making it a discrete distribution.
Additivity: Independent Poisson counts with parameters λ1 and λ2 combine to a Poisson with parameter λ1+λ2.
Relation to processes: Counts over disjoint intervals are independent in a Poisson process; inter-arrival times are exponentially distributed.

$\lambda$

Origins and relations

The name honors the mathematician Siméon Denis Poisson. One way to obtain the Poisson distribution is as a limit of the binomial distribution when the number of trials grows large while the expected number of successes remains fixed. For large λ, the Poisson can be approximated by a normal distribution with mean and variance λ; for small λ it models rare events more naturally than the normal. Its links to the exponential distribution and to processes with independent increments make it fundamental in the theory of stochastic processes. $X$

Applications, estimation and caveats

The Poisson model is widely used to count events such as calls to a call center, arrivals at a queue, rare defects in manufacturing, or occurrences of certain biological events. Parameter estimation is straightforward: the maximum likelihood estimator for λ is the sample mean of observed counts. Practical checks include testing for overdispersion (observed variance greater than the mean) which may indicate that a Poisson model is inadequate; alternatives include the negative binomial or mixture models. $P(X=x)={\frac {e^{-\lambda }\lambda ^{x}}{x!}}$

Useful distinctions and notes

Important assumptions are independence of events, a constant rate λ over the observation window, and the exclusion of simultaneous occurrences in an infinitesimally small interval. Violations of these assumptions—such as clustering, seasonality, or nonstationary rates—require extended models. For computational tools and further reading about derivations, proofs, and applications, see introductory probability texts and applied statistics resources. $x=0,1,2,\ldots$ For technical terms, you can follow related entries on probability distributions and on the lambda parameter.