Poisson Distribution Calculator

The Poisson distribution models the number of independent events that happen in a fixed interval when they occur at a known constant average rate λ. The random variable X counts the events. The chance of getting exactly k of them is given by the Poisson probability mass function (NIST §1.3.6.6.19):

P(X = k) = e^(−λ) · λᵏ / k!

Here λᵏ is the rate raised to the number of events, k! = k·(k−1)···1 is k factorial, and e^(−λ) normalises the whole distribution so the probabilities sum to 1. The cumulative questions are all sums of this one formula:

• At most k: P(X ≤ k) = Σ P(X = i), i = 0…k
• Less than k: P(X < k) = P(X ≤ k − 1)
• At least k: P(X ≥ k) = 1 − P(X ≤ k − 1)
• More than k: P(X > k) = 1 − P(X ≤ k)
• Between: P(k ≤ X ≤ k₂) = P(X ≤ k₂) − P(X ≤ k − 1)

To stay exact for a large rate, the calculator evaluates every term in log-space using a Lanczos log-gamma (DLMF §5), since k! = Γ(k+1): ln P(X=k) = −λ + k·ln λ − lnΓ(k + 1), then P = exp(ln P). This avoids the overflow that a direct λᵏ or k! would hit past a couple of hundred events. Cumulative sums run from k = 0 and stop once they are past the peak and the remaining terms are negligible, and P(X ≥ k) is taken as the complement 1 − P(X ≤ k − 1). As an independent check, the tool also builds the CDF from the recurrence P(i) = P(i−1)·λ/i starting at P(0) = e^(−λ) and shows the two methods agree.

The distribution summary uses the defining Poisson identity that the mean and variance are equal: μ = λ, σ² = λ and so σ = √λ, with the mode at ⌊λ⌋ (two adjacent modes, λ−1 and λ, when λ is a whole number), all per Wolfram MathWorld. When λ ≥ 10 the shape is close enough to a bell curve that the normal approximation N(λ, λ) is reasonable, and the tool flags this with a continuity-correction reminder. For a large number of rare trials, Poisson(np) also approximates the binomial B(n, p) — see the Binomial Distribution Calculator for the discrete sibling.

• NIST/SEMATECH e-Handbook §1.3.6.6.19 — Poisson Distribution (PMF, CDF, mean = variance = λ)
• Wolfram MathWorld — Poisson Distribution (PMF, CDF, mode ⌊λ⌋)
• NIST Digital Library of Mathematical Functions §5 — Gamma function (log-gamma basis for k!)

The formulas here were last cross-checked against these sources on 2026-07-09. They are mathematical constants, not policy, so they do not change with the calendar.