Poisson Distribution Calculator
The probability of a given number of events in a fixed interval, when events happen independently at a constant average rate.
The Poisson distribution answers "how many will happen?" when events arrive independently at a steady average rate λ over some interval — calls to a helpdesk in an hour, goals in a match, faults per kilometre of cable.
How the poisson distribution calculator works
The Poisson distribution answers "how many will happen?" when events arrive independently at a steady average rate λ over some interval — calls to a helpdesk in an hour, goals in a match, faults per kilometre of cable. The probability of exactly k events is e^−λ λᵏ ÷ k!.
It has an unusual property: the mean and the variance are both λ, so the spread is fixed by the rate. If your data has far more variation than its mean, it is over-dispersed and Poisson is the wrong model.
Formula: P(X = k) = e^−λ λᵏ ÷ k!
Worked examples
| Inputs | P(X = k) | Note |
|---|---|---|
| 6 events when the average is 4 | 10.4196% | 10.42% |
| No events when the average is 2 | 13.5335% | 13.53% — that is e⁻² |
| Exactly the average | 22.4042% | 22.4% |
FAQFrequently asked questions
When should I use a Poisson distribution?
When counting events that occur independently at a steady average rate over an interval — arrivals, faults, goals.
What is λ?
The average number of events expected in the interval. It is both the mean and the variance.
What is the probability of no events?
e raised to minus λ. With an average of 2, that is 13.5%.
How does it relate to the binomial?
It is the limit of the binomial when trials are many and the probability tiny, with n × p held at λ.
What if my data varies more than the mean?
Then it is over-dispersed and Poisson is the wrong model — a negative binomial usually fits better.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.