Poisson Distribution Calculator

In short The Poisson distribution answers "how likely is exactly k events in a period, when the average is λ". With an average of 3 arrivals an hour, exactly 3 happen 22.40% of the time and at least one happens 95.02% of the time. The formula is e^−λ × λᵏ ÷ k!, and it has a signature worth knowing: the mean and the variance are BOTH λ. That is the quickest test of whether your data is really Poisson — compute the mean and the variance of your counts, and if they are far apart it is not, whatever the histogram looks like. The standard deviation is √λ, which means counts get relatively less variable as the rate grows: ten times the rate gives only about three times the spread, so busy periods look steadier than quiet ones even when nothing about the process has changed. Poisson is what you get from a binomial with many trials and a small chance each, holding the average fixed, which is why rare events in large populations end up Poisson-shaped whatever the underlying mechanism. It models arrivals at a queue, radioactive decays, typos per page and goals in a match. The assumption that fails most often is independence: one delayed bus makes the next one late, outages cluster, and website traffic arrives in bursts — and when events cluster, the real tail is fatter than this page shows.