the Poisson distribution

The Poisson distribution counts how many times a rare, scattered event happens in a fixed window of time, space, or opportunity, when the events occur independently at a steady average rate. How many emails arrive in an hour? How many typos on a page? How many cars pass a sensor per minute? How many radioactive decays per second? In each case there is no natural upper limit n on the count — it could in principle be 0, 1, 2, and on up — and the whole distribution is governed by a single number, lambda, the mean number of events per window. We write X ~ Poisson(lambda).

Its pmf is P(X = k) = e^(-lambda) lambda^k / k! for k = 0, 1, 2, ... The e^(-lambda) is a normalizing constant that makes the probabilities sum to 1 (it is the famous series for e^lambda running in reverse), lambda^k grows with the count, and dividing by the factorial k! tames that growth. A striking and useful fact is that the mean and the variance are the same number: E[X] = lambda and Var(X) = lambda. So if you ever see count data whose sample mean and sample variance are roughly equal, a Poisson model is a natural first guess; if the variance is much bigger (overdispersion), you reach instead for the negative binomial.

Two stories justify the Poisson. As a limit, it is what the binomial becomes when you have an enormous number of trials n, each with a tiny success probability p, holding the product np = lambda fixed — the law of rare events. As a process, it is the count of arrivals in a Poisson process, where events drop independently and uniformly in time at rate lambda. Both pictures explain why it shows up everywhere from queueing and insurance claims to genetics and astronomy.