Poisson Distribution Calculator (with Graph)
Find Poisson probabilities for the number of events in an interval.
When the Poisson distribution applies
The Poisson distribution counts how many times an event happens in a fixed interval of time or space, when the events occur independently and at a constant average rate λ. Typical examples are calls to a help desk per hour, typos per page, cars passing a point per minute, or customers arriving per day. Unlike the binomial, there is no fixed number of trials, and the count can in principle be any whole number.
The formula
P(X = k) = e⁻λ × λᵏ ÷ k!. Both the mean and the variance equal λ, so the standard deviation is √λ. If the rate is quoted for a different period, scale it first: 3 calls an hour is 6 calls in 2 hours.
Worked examples (λ = 3 calls an hour)
| Question | Answer |
|---|---|
| Exactly 2 calls in an hour | 0.2240 |
| At most 4 calls | 0.8153 |
| At least 5 calls | 0.1847 |
| Between 1 and 3 calls | 0.5974 |
| No calls at all | 0.0498 |
The graph under the result highlights the event you chose, and has Download PNG and Download SVG buttons for reports and slides. A table lists P(X = k), P(X ≤ k) and P(X ≥ k) for the values around the mean.
Poisson and other distributions
- The Poisson is the limit of the binomial when n is large and p is small, with λ = n × p. For example, 500 items with a 0.6% defect rate has a mean of 3.
- The time between Poisson events follows the exponential distribution.
- When the variance is much larger than the mean (overdispersion), the Poisson is not a good fit.
How to cite this page
CalcPulse. (2026). Poisson Distribution Calculator. https://calcpulse.com/poisson-distribution-calculator/
Frequently asked questions
What is a Poisson distribution?
The distribution of the number of events in a fixed interval, when events happen independently at a constant average rate.
How do I calculate a Poisson probability?
Use P(X = k) = e⁻λ × λᵏ ÷ k!. For cumulative probabilities, add the individual probabilities.
What are the mean and variance of a Poisson distribution?
Both are equal to λ.
How do I change the time period?
Multiply λ by the length of the period. If calls average 3 an hour, use λ = 6 for 2 hours.
When is a Poisson distribution a good model?
When events are rare, independent and happen at a steady average rate. If the variance is much larger than the mean, it is not.
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