Poisson Distribution Calculator
Find the probability of a number of events in a fixed interval.
What the Poisson Distribution Calculator does
The Poisson distribution models the count of events in a fixed interval when they occur independently at a constant average rate — customer arrivals, server requests, equipment failures or typographical errors per page.
Formula
P(X = k) = (λᵏ × e⁻λ) ÷ k!Mean = Variance = λ
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Average events per interval (λ) | number | Yes | Accepts more than 0. |
| Number of events (k) | number | Yes | Accepts 0 or more. |
How to use it
- Enter Average events per interval (λ) and Number of events (k).
- Select Calculate.
Worked example
A call centre receives an average of 3 calls per minute — what is the chance of exactly 2 in a given minute?
- Lambda
- 3
- k
- 2
(3² × e⁻³) ÷ 2! = 0.2240, about 22.4%.
Reading the result
- "Variance = mean" is a testable signature of this model — the FAQ's overdispersion question hinges on checking that equality against real data, not just accepting the formula's output.
- For a small λ, "P(zero events)" is already meaningfully large — a low-rate process has a real chance of a quiet interval, not a near-certain "some events" outcome.
Assumptions and limitations
- This assumes a genuinely constant average rate over the interval — a period with an internal spike (like a lunch-hour rush) does not have one true λ, and blending it into a single average understates the probability of a burst during the busy sub-period.
- Events are assumed independent — the FAQ's overdispersion case often traces back to this failing, such as one event triggering a related second one.
Common mistakes
- Applying Poisson to a count of successes out of a known, fixed number of trials — the FAQ points to binomial for that case instead.
- Using a rate estimated from a short observation window as if it were the true long-run λ, without accounting for that rate's own sampling uncertainty.
Frequently asked questions
When should I use Poisson instead of binomial?
When there is no fixed number of trials — you are counting occurrences in an interval rather than successes out of n attempts.
What if the variance does not equal the mean?
That is overdispersion, and it means Poisson is the wrong model. The negative binomial distribution handles it better.