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A discrete random variable is subject to the generalized Poisson distribution with the parameters (event rate) and , if they are the
probabilities
owns. If one sets , the result is the usual Poisson distribution for the expected value .
properties
The variance is always at least as large as the expected value (for even greater). This property is called About dispersion ( English over-dispersion ).
For the generalized Poisson distribution, recursions for the sum distribution are known, as they are also known from the Panjer distribution .
For many use cases the implicit definition of the generalized Poisson distribution is sufficient.