Poisson distribution

The Poisson distribution models the number of events in a fixed interval when events occur independently at a constant average rate.

P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}, \quad k = 0, 1, 2, \dots

pmf

$\lambda$: expected number of events in the interval (rate × length of interval)
$k$: observed number of events

\E(X) = \lambda, \qquad \Var(X) = \lambda

Mean equals variance

$X$: a count, $X \sim \text{Poisson}(\lambda)$

When to use

Counts of events per unit of time, space or exposure: calls per hour, defects per metre, claims per month. It also approximates the binomial when $n$ is large and $p$ small ( $\lambda = np$ ).

Assumptions

Events occur independently, one at a time, at a constant rate; the count in one interval does not affect another. Rates scale with exposure: 3 per hour means $\lambda = 6$ for two hours.

How to interpret

Overdispersion check: for Poisson data the variance should be close to the mean. A sample variance much larger than the mean suggests clustering or a varying rate, and Poisson-based intervals and control limits will be too narrow.
The distribution is right-skewed for small $\lambda$ and close to normal for large $\lambda$ .

Common mistakes

Forgetting to rescale $\lambda$ to the interval in the question.
Using Poisson for counts with a fixed maximum and a high success rate (use the binomial).
"More than 4" is $\text{sf}(4) = P(X > 4)$ ; "at least 4" is $\text{sf}(3)$ .

Example

Calls arrive at $\lambda = 3$ per 10 minutes. $P(X = 2) = 0.2240$ ; $P(X \le 2) = 0.4232$ ; $P(X > 4) = \text{sf}(4) = 0.1847$ .