How this calculator works
The expected count combines rate and exposure. For a rate of two events per hour over three hours, enter six. Mean and variance are equal under this model. Clustering, changing rates and dependence may cause real counts to vary more than a Poisson model allows. λ = 0 means no events with certainty.
Formula
P(X=k) = exp(-λ)λ^k/k!; E[X] = Var[X] = λ.
Worked example
For λ = 2, no events have probability exp(-2) = 0.1353353. Exactly two have probability 2exp(-2) = 0.2706706.
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Frequently asked questions
Is λ the hourly rate?
Only when the window is one hour. Multiply rate by the window length first.
Can λ be fractional?
Yes. Expected counts need not be whole, although realised event counts must be.
What does the upper tail exclude?
P(X > x) excludes x. For at least k events, use one minus the cumulative probability at k - 1.