Poisson Distribution Calculator
For counts of events in a fixed time or space, such as arrivals per hour or defects per batch.
P(X = k)
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P(X ≤ k) —
P(X < k) —
P(X ≥ k) —
P(X > k) —
Standard deviation —
Reading this —
The formula
P(X = k) = λ^k e^(−λ) ÷ k!
The mean and variance both equal λ. The cumulative figures use the regularised incomplete gamma function.
What to watch out for
The rate must match the interval. If events average 6 an hour and you want the chance over 30 minutes, use λ = 3.
Frequently Asked Questions
What is a worked example?
With λ = 3, P(X = 2) is 0.22404, and P(X = 0) is 0.049787.
How does it relate to the binomial?
It is the limit of a binomial with many trials and a small chance on each, with λ = np.