Statistics · Statistical distributions
Chapter 1 · 3
The idea
The Poisson distribution
Modelling the number of random events in a fixed interval — when the Poisson model is valid, P(X = x) = e^{−λ}λ^x/x!, mean = variance = λ, scaling λ to the length of the interval, adding independent Poissons, and using Poisson as an approximation to the binomial when n is large and p is small.
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Statistics · Statistical distributions
The Poisson distribution
Modelling the number of random events in a fixed interval — when the Poisson model is valid, P(X = x) = e^{−λ}λ^x/x!, mean = variance = λ, scaling λ to the length of the interval, adding independent Poissons, and using Poisson as an approximation to the binomial when n is large and p is small.
Why it works
Counts with no number of trials
Some counts have no natural "number of trials": calls arriving at a switchboard, flaws in a metre of cloth, misprints on a page. There is no to put in a binomial — events just happen, one at a time, at some average rate. The Poisson distribution is the model for exactly this situation. If events occur- singly (two can't land at exactly the same instant),
- independently (one arriving tells you nothing about the next), and
- at a constant average rate.
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