Poisson Probability Calculator

Poisson distribution: probability of exactly k events

The Poisson distribution tells you how likely you are to see exactly k events in a fixed interval, assuming those events happen independently at a steady average rate λ. The formula is P(X = k) = e^(−λ)·λ^k / k!. One thing that sets it apart is that its mean and variance are both equal to λ. It got its fame from Ladislaus Bortkiewicz, who in 1898 used it to study how many Prussian cavalrymen were killed each year by horse kicks. Take λ = 3 and k = 2: P(X = 2) = e^(−3)·9 / 2 ≈ 0.224, so roughly a 22.4% chance of seeing exactly 2 events. And when n grows toward infinity while p shrinks toward zero with n·p = λ, the binomial collapses into the Poisson, which is why it fits rare events in large populations so well.

Applications

You'll find it behind queueing models like M/M/1 in call centers and packet networks. It shows up in radioactive decay counts, in counting defects per unit on a factory line, in how customers trickle into a store, in mutation counts across genomes, and even in goals per football match (somewhere near 1.5 per side in the Premier League).

FAQ

What is k!? That's the factorial: 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24. If you ever need it for non-integer arguments the gamma function takes over, but here k has to be a non-negative whole number.

What if variance ≫ mean? Then your data is overdispersed and Poisson isn't the right fit. Reach for the negative binomial instead, since it carries an extra dispersion parameter.

Cumulative probability? Add up the individual terms: P(X ≤ k) = Σ P(X = i) for i = 0..k. When λ gets large, the normal approximation N(λ, λ) does the job.

How do I estimate λ? The maximum-likelihood estimator turns out to be nothing more than the sample mean: λ̂ = x̄.