About this tool
The Poisson Distribution Calculator gives the probability of a number of events k occurring in a fixed interval when they happen independently at a constant average rate λ (lambda). Enter the mean rate and a count and it returns the exact and cumulative probabilities live, all computed in your browser.
The probability mass function is P(X = k) = λᵏ·e^(−λ) / k!. To stay numerically stable for large counts the tool evaluates it in log space as exp(−λ + k·ln λ − ln k!), using a Lanczos approximation for the log-factorial. It also sums the mass function to give the cumulative P(X ≤ k), the strict P(X < k), and the upper tail P(X ≥ k). A defining property of the Poisson distribution is that its mean and variance are both equal to λ, so the standard deviation is √λ.
Use it for arrivals per hour, defects per batch, calls per minute, decay counts and any rare-event process where the count is a non-negative integer. The rate λ must be greater than zero and k must be a whole number of zero or more.