Poisson Distribution: The Law of Rare Events

The binomial distribution assumes a known, fixed $n$ (the number of trials) and a not-too-small probability of success $p$. Many real business situations do not fit that mould — there is no fixed, countable number of 'trials' for the number of accidents at a factory in a day, the number of customer complaints an airline receives in an hour, or the number of defective items in a very large production batch where a defect is genuinely rare. These are situations where events happen randomly and independently over a continuous interval (of time, area or volume) at a known average rate.

The Poisson distribution is the theoretical distribution for exactly this situation. It can also be derived as the limiting form of the binomial distribution as the number of trials $n$ becomes very large and the probability of success $p$ becomes very small, in such a way that their product $np$ settles down to a constant, called $\lambda$ (lambda) — the average number of occurrences per interval. This is why the Poisson distribution is often used as a convenient approximation to a binomial distribution whenever $n$ is large (commonly $n \ge 20$) and $p$ is small (commonly $p \le 0.05$).

If $X$ is the number of occurrences of a rare event in a fixed interval, and the average rate of occurrence is $\lambda$, then:

A distinctive property of the Poisson distribution, unlike the binomial, is that its mean and variance are equal, both equal to $\lambda$:

$E(X) = \lambda \qquad\qquad Var(X) = \lambda$ …