Poisson Distribution Calculator

The Poisson distribution models the number of times an event occurs in a fixed interval (time, area, or volume) when those events are independent and the average rate is constant. It is parameterized by a single number, lambda (λ), which is both the mean and the variance. Real-world examples: calls arriving at a help desk per hour, typos per page, photons reaching a sensor per second.

P(X = k) = e^(−λ) × λ^k ÷ k!. e is Euler's number (≈ 2.71828). λ is the average rate. k is the integer count of events. The calculator uses an iterative form for stability: P(0) = e^(−λ), then P(i) = P(i − 1) × λ ÷ i. This avoids computing large factorials directly.

Three conditions. (1) Events occur independently of each other. (2) The average rate λ is roughly constant over the interval. (3) Two events cannot occur at the exact same instant. Common violations: rush-hour traffic with bursty arrivals, contagious infections where one case raises the probability of the next, or any process with a hard cap on events.

λ (lambda) is the expected number of events in the interval you care about. If a help desk averages 12 calls per hour and you want the probability for a 30-minute window, use λ = 6 for that window. Make sure λ and k refer to the same fixed interval. The mean equals λ; the variance also equals λ, which is a distinctive property of the Poisson distribution.

P(X = k) is the chance of exactly k events. P(X ≤ k) is the chance of k or fewer events (cumulative up through k). P(X ≥ k) is the chance of k or more events. The calculator computes the cumulative form by summing the per-i probabilities iteratively, and uses 1 − P(X ≤ k − 1) for the at-least case to keep the result numerically stable.

The Poisson distribution is the limiting case of the binomial when n is large, p is small, and n × p stays finite. Setting λ = n × p and using Poisson gives a good approximation when np stays roughly constant as n grows. In practice, for n above 50 and p below 0.05, Poisson with λ = np is often used instead of binomial because the math is easier.