Poisson distribution calculator
Fits counts of random events such as calls per hour or customers arriving in 10 minutes.
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Processing and privacy: The result is calculated inside your browser and your data is not sent to any server.
How to use it
- Enter average rate λ and number of events k.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
P(X = k) = e^(-λ) × λ^k ÷ k!. Cumulative values are sums of these terms. Mean and variance both equal λ.
Example
Inputs: Average rate λ = 3, Number of events k = 2. Result: P(X = k): 0.2240418077; P(X < k): 0.1991482735; P(X ≤ k): 0.4231900811.
Limits and caveats
- Assumes independent events at a constant average rate over the interval.
- Use λ for the same interval you ask about: 6 per hour is 3 per half hour.
Questions about this tool
What does λ mean?
The average number of events in the interval, like 3 calls per hour. It can be a decimal such as 2.5.
What is the chance of no events at all?
P(X = 0) = e^(-λ). For λ = 1 it is about 0.3679.
When does Poisson approximate the binomial?
When n is large and p is small, λ = n × p gives a good approximation of the binomial.