Counting events when only the rate is known
Calls per hour, typos per page, arrivals per minute: the Poisson distribution prices counts in a fixed window when events arrive independently at a steady average rate λ. There is no n and no p; the rate is the whole input, and remarkably the mean and the variance of the count are both exactly λ.
A help desk averages λ = 4 calls per hour. Exactly 2 calls: e−4 × 4² ÷ 2! = 0.1465. Two or fewer: 0.2381, nearly a quarter of hours are that quiet. Five or more: 0.3712. The busiest-looking hours need no explanation beyond the distribution itself; Poisson variation is lumpy by nature.
Scaling the window
λ scales with the window: 4 per hour is 32 per 8-hour day and 0.0667 per minute. Always convert the rate to the window your question asks about before calculating: the most common Poisson error in practice is a correct calculation on the wrong window.
Poisson assumes events do not cluster or repel, no aftershocks, no batch arrivals. Counts that bunch (calls after an outage announcement) spread wider than λ, and the negative binomial is the model that admits it.
Frequently asked questions
Where does Poisson come from?
It is the binomial's limit: infinitely many microscopic trials, each with a tiny success chance, holding the average count np = λ fixed. Rare events in many opportunities, exactly the Poisson's habitat.