Poisson Distribution Calculator

Compute probabilities for a Poisson distribution with mean rate λ: the probability of exactly k events, of at most / at least k events, the cumulative distribution, and the mean, variance, and mode. A table shows the probability mass and cumulative probability around k. Computation is done in log-space so it stays accurate for large λ and k. Pairs with the Binomial and Normal distribution tools. Everything runs locally in your browser.

λ (mean rate, > 0) k (count)

Probabilities at k

Distribution table

k	P(X = k)	P(X ≤ k)

Poisson PMF: P(X = k) = e^−λ λ^k / k!. Cumulative P(X ≤ k) = Σ P(X = i) for i = 0…k. Computed in log-space (ln P = −λ + k·ln λ − ln(k!)) to avoid factorial overflow, so large λ (e.g. 100) and large k are handled accurately. The mean and variance of a Poisson variable both equal λ; the mode is ⌊λ⌋ (two adjacent modes when λ is a whole number). The table covers a window around the mean (capped at 60 rows). Everything runs locally — nothing leaves your browser.