6.1.1—Poisson probabilities

If events occur independently at constant mean rate λ per interval, X~Po(λ) and P(X=r)=e^{−λ}λ^r/r!.

Match λ to the interval length, check that events are countable and rare enough for the model, and use complements for “at least one” questions.

If a call centre averages 3 calls per minute, P(2 calls in one minute)=e^{−3}3²/2.

Changing the interval changes λ proportionally; it is not a universal parameter for every time window.