The Poisson Distribution

How many buses pull up at your stop in the next hour? How many emails land in your inbox before lunch? How many shooting stars streak across the sky in a ten-minute wish-making window? Each of these counts rare, random events happening in a fixed stretch of time — and the same tidy piece of mathematics describes them all. It is called the Poisson distribution, after Siméon Denis Poisson.

The setup is always the same. Events happen one at a time, they are independent (one bus arriving tells you nothing about the next), and they occur at a constant average rate which we call \lambda (the Greek letter "lambda"). Fix an interval — an hour, a page, a square metre — and let X be the number of events in it. We write

X \sim \mathrm{Poisson}(\lambda),

and read it "X is distributed as Poisson with rate \lambda". The single number \lambda is the expected count for that interval — the average you'd see if you watched a great many identical intervals go by.

What do a call centre, a proof-reader, and a Geiger counter have in common? Calls to the centre per hour, typos per page, radioactive decays per second — each is a swarm of rare little events, each one unlikely at any given instant, but with so many chances to happen that a handful sneak through in every interval. That "many chances, each tiny" flavour is exactly what the Poisson distribution captures. Whenever you can point at a window of time or space and ask "how many landed in there?", Poisson is the natural first model to reach for.

The formula

The probability of seeing exactly k events in one interval is

P(X = k) = \frac{e^{-\lambda}\,\lambda^{k}}{k!}, \qquad k = 0, 1, 2, 3, \ldots
k ranges over all the non-negative whole numbers — 0, 1, 2, \ldots with no upper limit. In principle a very unlucky page could have a hundred typos.
Its expected value (mean) and its variance are both equal to \lambda: E(X) = \operatorname{Var}(X) = \lambda.

Every piece earns its place. The \lambda^{k} grows the more events you ask for; the k! underneath punishes long runs (there are lots of ways to arrange many events, but each becomes unlikely); and the e^{-\lambda} out front is exactly the constant that makes all the probabilities add up to 1 — because \sum_{k=0}^{\infty}\lambda^{k}/k! = e^{\lambda}, the famous series for e^{\lambda}.

Worked example 1: the corner shop

A small shop serves on average \lambda = 3 customers every 10 minutes. Treating arrivals as Poisson, what is the chance that exactly 5 customers walk in during the next ten-minute spell?

P(X = 5) = \frac{e^{-3}\,3^{5}}{5!} = \frac{e^{-3}\cdot 243}{120} \approx \frac{0.0498 \cdot 243}{120} \approx 0.101.

So about a 10\% chance. And the probability that nobody comes in — a rare quiet spell — is the simplest case of all, k = 0:

P(X = 0) = \frac{e^{-3}\,3^{0}}{0!} = e^{-3} \approx 0.0498,

using 3^{0} = 1 and 0! = 1. So roughly a 1-in-20 chance of a completely empty ten minutes.

Worked example 2: mean equals variance

The most surprising fact about the Poisson distribution is that its mean and variance are the same number, \lambda. The centre of the distribution and its spread are locked together: crank up the rate and the bump both slides right and widens by the same amount.

Take radioactive decays at \lambda = 4 per second. The average count is 4, and so is the variance, which means the standard deviation is \sqrt{4} = 2. A typical second lands roughly in the range 4 \pm 2. Because the standard deviation is \sqrt{\lambda}, the count becomes relatively steadier as \lambda grows: the wobble is \sqrt{\lambda} against a mean of \lambda, a smaller and smaller fraction.

This also changes the shape. For small \lambda the distribution is bunched at 0 and right-skewed (a long tail reaching to the right, since counts can't go below zero). For large \lambda it spreads out and becomes almost symmetric — a bell-like mound centred on \lambda.

See it: the shape of the distribution

The curve below traces P(X = k) for each count k. Drag the \lambda slider and watch the whole distribution respond: at small \lambda it is a lopsided spike hugging the left edge; as \lambda climbs, the peak marches right to sit near k = \lambda and the mound widens, drifting toward a symmetric bell.

Poisson is really the binomial distribution in disguise. Imagine chopping your hour into n tiny slots, so tiny that each holds at most one event, with a tiny success probability p per slot. The count of events is B(n, p). Now let the slots become infinitely fine — n \to \infty and p \to 0 — while holding the expected count np = \lambda fixed. The binomial formula collapses exactly onto \dfrac{e^{-\lambda}\lambda^{k}}{k!}. That is why Poisson is the law of many trials, each with a tiny chance.

Poisson counts events in a fixed interval at a constant rate. If the rate changes through the interval (a shop is busy at noon, dead at 3pm), a single \lambda won't do.
Its mean and variance are equal (both \lambda). If real data has a variance much bigger than its mean — called overdispersion — the Poisson model is the wrong tool, and you need something more flexible.
k can be any non-negative integer, with no upper limit. This is unlike the binomial, where the count can never exceed n.
\lambda must match the interval you're asking about. A rate of 3 per hour becomes \lambda = 1.5 for a half-hour and \lambda = 6 for two hours — always rescale first.

Once you can count events in time, the natural next question is how long you wait between them — that gap follows the exponential distribution, the Poisson's close cousin.