Poisson distribution

Calls arriving at a switchboard, typos per page or particles detected per second are counts of events that occur independently at a steady average rate. The parameter \(\lambda\) is the mean number of events in the interval considered and the probability of observing exactly \(k\) is \(e^{-\lambda}\lambda^k/k!\).

A distinctive property is that mean and variance coincide: \(E[X]=\operatorname{Var}(X)=\lambda\). If the rate is \(r\) events per unit, an interval of length \(t\) uses \(\lambda=rt\): 6 calls per hour mean \(\lambda=3\) in half an hour.

The Poisson arises as a limit of the binomial: when \(n\to\infty\) and \(p\to0\) with \(np=\lambda\) fixed, \(\binom nk p^k(1-p)^{n-k}\to e^{-\lambda}\lambda^k/k!\). That is why it approximates \(\mathrm{Bin}(200,0.01)\) well with \(\lambda=2\).

The sum of independent Poisson variables with parameters \(\lambda_1\) and \(\lambda_2\) is Poisson with parameter \(\lambda_1+\lambda_2\). A common mistake is to apply it when events cluster or the rate changes: then the variance exceeds the mean and the model does not fit.