The distribution of the number of events in a fixed interval when events happen independently at a constant average rate λ: P(k) = λᵏ e^(−λ) / k!. Its mean and its variance are both λ.
Card 1 of 244. Answer: The distribution of the number of events in a fixed interval when events happen independently at a constant average rate λ: P(k) = λᵏ e^(−λ) / k!. Its mean and its variance are both λ.
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The Poisson distribution models counts: calls per hour, typos per page, requests per second. It assumes events occur independently, at a constant average rate, and one at a time. Its single parameter λ is both the expected count and the variance, and the waiting time between events follows an exponential distribution.
Example: a support desk gets 4 calls an hour on average. The chance of no calls in an hour is e⁻⁴ ≈ 1.8%, and of exactly 4 calls is 4⁴e⁻⁴/4! ≈ 19.5%. The chance of 7 or fewer is about 95%, so staffing for 7 calls an hour covers roughly 19 hours in 20.
It is also the limit of the binomial when n is large and p small, with λ = np, which is why it fits rare events; Ladislaus Bortkiewicz famously fitted it to deaths by horse kick in the Prussian army in 1898. The usual follow-up is overdispersion: real counts often have a variance well above the mean, and then a negative binomial model fits better. Poisson regression, with a log link, is the standard model for count targets.
