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A discrete distribution defined on the non-negative integers that has the property that the mean is equal to the variance.

Overview

A discrete random variable $X$ has a Poisson distribution indexed by a parameter $\lambda$ if it has probability mass function

$$ P(X = x) = \frac{ \lambda^x e^{-\lambda} }{x!} \quad \text{for } x>0 $$

One property of the Poisson distribution is that $\mathrm{E}(X) = \mathrm{Var}(X) = \lambda$.

The Poisson distribution is used to model situations where there is a rate of occurrence associated with an event. For example, it used prominently in Physics to model "counting experiments" like the number of photons arriving at a telescope, or the number of radioactive counts recorded by a Geiger counter.

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