Poisson Cumulative Probability Calculator

About this calculator

The Poisson distribution models the number of times a rare, independent event happens in a fixed interval of time or space, given a known average rate λ (lambda). The probability of exactly k events is P(X = k) = e^(−λ) λ^k / k!, and the cumulative probability P(X ≤ k) — the probability of seeing k or fewer events — is the running sum of that formula from 0 up to k.

This distribution and its cumulative probability are used wherever counts of rare independent events need modeling: call centers forecasting how many calls will arrive in an hour, reliability engineers estimating how many failures a system will see in a given period, epidemiologists modeling disease case counts, and insurers pricing policies based on the expected number of claims. The cumulative version answers practical questions like ‘what is the chance of getting 10 or fewer calls this hour,’ rather than just the probability of exactly 10.

Enter a rate λ and a count k, and the calculator computes the cumulative Poisson probability P(X ≤ k), along with the exact probability at k and the distribution's mean.