Many real-world counts — calls hitting a switchboard, atoms decaying, typos on a
page, cars arriving at a junction — share a pattern: events happen independently,
at a constant average rate, and any two events are (in principle) unrelated. The
Poisson distribution says that if the long-run average count per
interval is λ (lambda), the probability of observing exactly
k events is P(k) = λᵏ · e⁻λ / k!. Remarkably, this single
number λ is enough to predict the whole shape of the distribution — including how
spread out it is, since for a Poisson process the variance always equals the mean.
k for that interval and spawns exactly k falling events at random times inside it — exactly how a true Poisson process behaves.k, computed directly from λ. As more intervals accumulate, the solid bars should hug the wireframe target more and more closely — the Law of Large Numbers in action.The Poisson distribution was first used in 1898 by Ladislaus Bortkiewicz to model the number of Prussian cavalrymen killed by horse kicks each year — a classic case of a rare, independent event counted over a fixed interval.
Rare, independent events fall from the sky at random moments and fly into a live 3D histogram — watch it converge on the theoretical Poisson curve as more intervals are simulated.
Each interval, the number of arriving events is drawn from a true Poisson process with rate λ. The solid bars track the observed frequency of each count; the wireframe bars show the exact theoretical probability P(k) = λᵏe⁻λ/k!.
Set the average rate λ and simulation speed, pick an event context for flavour, and watch counts accumulate. Toggle the theoretical overlay off to see the raw sample histogram, and reset to start a fresh run.
For a genuine Poisson process the variance always equals the mean — watch the sample variance stat drift toward the same value as the sample mean as intervals accumulate.