Poisson Distribution Calculator

When events happen independently, at a constant average rate, with no two occurring at exactly the same instant: arrivals, requests, failures, rare defects, decays. If events cluster (one triggers another) or the rate varies systematically, Poisson under- or over-states the spread. Its hallmark assumption is independence at a steady rate λ.

It's a defining mathematical property: both equal λ. This is a useful diagnostic — if your count data has variance much larger than its mean ('overdispersion', common in real data with clustering), plain Poisson underestimates the spikes and you need a negative binomial or quasi-Poisson model instead. Equal mean and variance is the assumption to check.

If requests arrive Poisson at rate λ, the distribution tells you how often demand exceeds any threshold — so you size capacity for, say, the 99th-percentile load, not the average. A system built for the mean fails constantly during normal random spikes. Poisson (and the queueing theory built on it) quantifies exactly how much headroom you need.

Poisson is the limit of the binomial when you have many trials (n large) each with a tiny success probability (p small), with λ = np. So for rare events in many opportunities — defects in a long production run, mutations across a genome — Poisson is the simpler, accurate approximation to the binomial, needing only the average rate rather than n and p separately.