Poisson distribution

The story

Imagine events that arrive at random but at a steady average rate: emails landing in an inbox, clicks on a Geiger counter. Chop the window into a huge number of tiny slices, each of which almost never contains an event, and the total count is a Binomial with enormous $n$ and tiny $p$ . In the limit you get the Poisson, with $P(X = k) = e^{-\lambda} \lambda^k / k!$ .

Its defining quirk is that the mean and the variance are both equal to $\lambda$ . Small $\lambda$ gives a right-skewed pile of bars near zero; as $\lambda$ grows the distribution becomes more symmetric and bell-shaped, with standard deviation $\sqrt{\lambda}$ .

Formulas

Support

k \in \{0, 1, 2, \ldots\}

PMF

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

CDF

F(k) = e^{-\lambda} \sum_{i=0}^{k} \frac{\lambda^i}{i!}

Expectation

\mathbb{E}[X] = \lambda

Variance

\operatorname{Var}(X) = \lambda

Mode

\lfloor \lambda \rfloor

Parameters

Parameter	What it controls
$\lambda$	The average number of events per window. It is both the mean and the variance, so larger $\lambda$ moves the distribution right and spreads it out.

Where it shows up

The number of calls reaching a call centre in a ten-minute window.
Typos per page in a long manuscript.
Radioactive decays detected per second.
Goals scored by a football team in a match.

Try this

Press Draw 1000 at $\lambda = 4$ and compare the sample mean and sample variance: both should land close to 4, the signature of a Poisson count.
Slide $\lambda$ from 0.5 up to 20 and watch the shape change from a lopsided pile at zero to an almost symmetric bell centred on 20 with $\sigma \approx 4.47$ .
Set $\lambda = 1$ and shade the interval from 0 to 0: the shaded probability is $e^{-1} \approx 0.368$ , the chance that nothing happens at all in a window where one event is expected.

Related distributions

It is the limit of the $\text{Binomial}(n, p)$ as $n \to \infty$ and $p \to 0$ with $np = \lambda$ fixed.
In a Poisson process with rate $\lambda$ , the waiting times between events are independent $\text{Exponential}(\lambda)$ variables.
The sum of independent Poisson variables is Poisson, with the rates added together.
For large $\lambda$ it is well approximated by a Normal with mean $\lambda$ and variance $\lambda$ .