Poisson Distribution Calculator
Find Poisson probabilities for the number of events in a fixed interval, given the average rate.
About the Poisson Distribution Calculator
The Poisson distribution models counts of rare, independent events in a fixed time or space: calls to a help desk per hour, typos per page, or goals in a match. Its single parameter λ is both the mean and the variance.
Two engineering exercises are built in: the Poisson approximation to a binomial with large n and small p (λ = np, compared with the exact value), and fitting a Poisson distribution to a frequency table (λ = the mean, then expected frequencies N·e^(−λ)λˣ/x! and the chi-square statistic).
The formula
P(X = k) = λᵏ e^(−λ) ÷ k!
Worked example
A shop averages 3 customers per 10 minutes
- P(exactly 5) = 3⁵e⁻³ ÷ 120
- = 0.1008
Questions people ask
When does Poisson apply?
When events occur independently at a constant average rate and two can't happen at exactly the same instant.
How is it related to the binomial?
It's the limit of the binomial when n is large and p is small, with λ = np.