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The Poisson distribution can be defined as the limiting case of the binomial distribution for but = const.
It thus describes the behaviour of a large number n
of independent experiments of which only a very small fraction pn is expected to yield events of a given type A.
As an example, n may be the number of radioactive nuclei in a source and p the probability for a nucleus to decay in a fixed interval of time.
The probability for X=k events of type A to occur is
The distribution has the following properties
If k events are observed, is an unbiased estimator of the single parameter . The variance of is also equal to ,
hence approximately equal to
A simple generator for random numbers taken from a Poisson distribution is obtained using this simple recipe: if is a sequence of random numbers with uniform distribution between zero and one, k is the first integer for which the product .
Rudolf K. Bock, 7 April 1998