Definition
The Poisson distribution describes the number of events occurring in a fixed interval (of time, space, or some other dimension) when events happen independently and at a constant average rate lambda. The probability of exactly k events is P(X = k) = e^(-lambda) × lambda^k / k!.
Both the mean and variance of a Poisson distribution equal lambda — a distinctive property that makes Poisson easy to recognise in data.
Why it matters
How it works
To use a Poisson, estimate lambda — the expected number of events in your chosen interval. Then probabilities of specific counts follow from the formula. For example, if a website receives an average of 3 visits per minute, the probability of receiving exactly 5 visits in a given minute is e^(-3) × 3^5 / 5! ≈ 0.10.
For larger lambda, the Poisson is approximately normal with mean and variance both lambda, allowing the same normal-approximation tools to be reused. For very small lambda, almost all probability concentrates at 0 — most intervals see no events at all.