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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