Statistics Calculators

Poisson distribution calculator

Compute the probability of exactly k events given an expected rate λ — Poisson PMF for small non-negative k.

Poisson Distribution Calculator

Table of contents

◦Poisson Distribution Calculator
◦Formula
◦How to use
◦Worked example
◦FAQ

Poisson Distribution Calculator

The Poisson distribution models the number of events occurring in a fixed interval when each event happens independently at a known average rate λ.

This calculator returns P(X = k) — the probability of exactly k events occurring — for any non-negative integer k and any positive expected rate λ.

Formula

P(X = k) = e<sup>−λ</sup> · λ<sup>k</sup> / k!

Where:

  • λ (lambda) is the expected number of events in the interval
  • k is the specific event count you're evaluating
  • k! is the factorial of k

Because our formula engine doesn't have a native factorial, we use Stirling's approximation for k!:

k! ≈ √(2πk) · (k/e)<sup>k</sup>

Stirling's approximation is accurate to under 1% for k ≥ 5 and improves rapidly. For k = 0 and k = 1 we substitute 0.5 inside the approximation to keep the formula numerically stable without affecting the very small error.

How to use

  1. Enter λ — the expected event rate. Examples: 3 calls per hour at a help-desk, 1.5 goals per match, 0.8 buses per minute at a stop.
  2. Enter k — the specific number of events you want the probability for.
  3. The probability appears instantly.

Worked example

A call center averages 3 calls per hour (λ = 3). What's the probability of receiving exactly 5 calls in a given hour (k = 5)?

P(X = 5) = e<sup>−3</sup> · 3<sup>5</sup> / 5!

= 0.0498 · 243 / 120

≈ 0.1008 (10.08%)

FAQ

When should I use Poisson vs. Binomial?

Use Poisson when events happen continuously and independently at a known average rate. Use Binomial when there's a fixed number of trials and each has the same probability of success. Poisson is the limiting case of the Binomial as n → ∞.

How accurate is the Stirling approximation here?

For k = 5 the error is ~1.6%. For k ≥ 10 it's under 1%. For k = 0 and k = 1, the substitution keeps the formula well-defined; the absolute error is small because exact k! at those points is tiny (1 and 1 respectively, and we're computing a probability that's between 0 and 1).

What if my k is very large (k > 20)?

For large k, the Poisson PMF approaches a Normal distribution with mean λ and variance λ. Use a normal-distribution approximation for k > 20.