Many random variables count how many events occur during a fixed amount of time, distance, area or other exposure. When events occur independently at an approximately constant average rate, a Poisson distribution is a common model.

If the expected count over the chosen interval is $\lambda>0$, then

$$X\sim\operatorname{Poisson}(\lambda)$$

has probability mass function

$$P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!},\qquad k=0,1,2,\ldots$$

Example

Suppose a help desk receives an average of $3$ calls per hour and a Poisson model is appropriate. Then the probability of exactly two calls in an hour is

$$P(X=2)=e^{-3}\frac{3^2}{2!}.$$

Mean and variance

A distinctive property of the Poisson distribution is

$$E[X]=\lambda,$$

$$\operatorname{Var}(X)=\lambda.$$

Thus the single parameter $\lambda$ controls both center and spread.

Scaling the interval

If the average rate is $r$ events per unit exposure and the observed exposure is $t$, then

$$\lambda=rt.$$

Doubling the time interval therefore doubles the expected count when the rate remains constant.

Model assumptions

A Poisson model is plausible when events occur independently, the underlying rate is approximately constant and simultaneous events are negligible at sufficiently small scales.

Clustering, strong time variation or a fixed maximum number of possible events can make the model inappropriate.