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Mihsaba›Poisson Distribution Calculator

Poisson Distribution Calculator

Calculate exact, at-most and at-least Poisson probabilities from mean rate λ and integer count k, plus mean, variance and standard deviation.

How to use

The Poisson distribution models a count of events in a fixed time or space interval when the expected mean λ is known and a constant-rate, independent-event model is appropriate.

  1. Enter mean λ from 0 to 1000.
  2. Enter integer event count k from 0 to 10,000.
  3. Calculate P(X=k), P(X≤k), P(X≥k), mean, variance and standard deviation.

How it is calculated

P(X=k)=e⁻λ λᵏ÷k!; μ=σ²=λ

Example

Example: with λ=4 and k=2, P(X=2)≈14.65% and P(X≤2)≈23.81%.

Important notes

The calculator uses numerically stable cumulative-probability formulas instead of naively summing a large number of terms. Whether the result is a good real-world model depends on the Poisson assumptions fitting the data.

Frequently asked questions

Exact versus at least?

P(X=k) is one count, while P(X≥k) includes k and every larger count.

What happens at λ=0?

Then X=0 with probability 1 and every positive count has probability 0.

Is this calculator free?

Yes. It is free to use and requires no account.