The Poisson distribution: modeling event counts in time or space
The Poisson distribution is a classic model for counting how many times an event occurs in a fixed interval,
such as calls arriving per hour, flaws appearing per meter of wire, or customers entering a store per minute.
We write \(X\sim\mathrm{Pois}(\lambda)\), where the parameter \(\lambda\) is the average rate of events in that interval.
For example, \(\lambda=2.5\) could mean an average of 2.5 calls per hour.
When is a Poisson model reasonable?
A standard set of assumptions behind the Poisson model is that events happen independently and “uniformly” through the interval.
Informally, this means the probability of an event in a very small sub-interval is proportional to the length of that sub-interval,
and the chance of two or more events in that tiny sub-interval is negligible.
Real data will not match these assumptions perfectly, but the Poisson distribution is often a good starting point for rare or
randomly scattered events.
PMF: probability of exactly \(k\) events
The probability of seeing exactly \(k\) events is given by the probability mass function (PMF):
\[
P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!},\qquad k=0,1,2,\dots
\]
The factor \(e^{-\lambda}\) controls the overall scale, \(\lambda^k\) grows with \(k\), and \(k!\) grows very quickly,
so probabilities eventually decrease for large \(k\). In practice, calculators evaluate this formula using stable methods
(often with \(\log(k!)\)) to avoid overflow when \(k\) is large.
CDF: probability of at most \(k\) events
Many questions are cumulative, like “at most 3 calls” or “no more than 1 defect”.
The cumulative distribution function (CDF) is a sum of PMF values:
\[
P(X\le k)=\sum_{x=0}^{k} e^{-\lambda}\frac{\lambda^x}{x!}.
\]
As \(k\) increases, the CDF rises from 0 toward 1. Visualizing both PMF bars and the CDF curve helps you see how probability
accumulates across counts.
Mean and variance
A special feature of the Poisson distribution is that its mean and variance are both equal to \(\lambda\):
\[
\mathbb E[X]=\lambda,\qquad \mathrm{Var}(X)=\lambda.
\]
So the typical size of fluctuations grows like \(\sqrt{\lambda}\). When \(\lambda\) is larger, the distribution spreads out more,
and the Poisson shape begins to resemble a bell curve.
University note: beyond the basic model
In more advanced settings, the Poisson process can be extended to non-constant rates (inhomogeneous Poisson processes)
or to compound Poisson models, where each event contributes a random “size” (useful in insurance and queueing theory).
The core idea remains the same: start with a rate model and compute probabilities of counts.
How to use this tool
Enter \(\lambda\) and \(k\), then click Calculate to compute \(P(X=k)\) and (optionally) \(P(X\le k)\) with step-by-step work.
The interactive canvas shows PMF bars on top and the CDF line below. You can drag to pan, wheel to zoom, and press Play
to sweep the highlighted \(k\) across the visible window and observe how probabilities change.
