Poisson Distribution

Rare-event counts: the pmf, mean and variance λ, and the binomial limit.

The idea

The Poisson distribution describes counts: how many times an event occurs in a fixed window, when the opportunities for it are many and each is individually unlikely — calls reaching a switchboard in an hour, typos on a page, particles registering in a detector in a second.

We could model such a count as $n$ independent trials, each succeeding with probability $p$, and use the binomial distribution. The obstacle is that $n$ is enormous and not even well defined: there is no natural number of separate chances for a call to arrive during an hour. What we can measure is the average count, the product $np$.

So hold $\lambda = np$ fixed, let $n$ grow without bound with $p$ shrinking to match, and take the limit of the binomial. The result is the Poisson distribution with rate $\lambda$: a random variable $X$ taking the values $k = 0, 1, 2, \ldots$ with $\mathbb{P}(X = k) = e^{-\lambda}\,\frac{\lambda^{k}}{k!}.$ The unmeasurable $n$ has disappeared, and the one remaining parameter $\lambda$ is the mean — it was the average count all along — and, less obviously, also the variance.

Ways to work on it

• Walkthrough. The pmf, mean, and the rare-event (binomial) origin.
• Practice. Evaluate the Poisson pmf at a small count.
• Hardest. Tail probability via the complement, and the binomial-limit rationale.

Not sure where to start? Take the ten-question placement test.