Generalized Poisson Distribution

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The generalized Poisson distribution is a probability distribution and can therefore be assigned to the mathematical sub-area of stochastics . It is a univariate discrete probability distribution on the natural numbers, which is mainly used in actuarial mathematics . Compared to the Poisson distribution , it has two parameters and is therefore much more flexible than this.

definition

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.

Expected value

The expected value results in

.

Variance

For the variance one gets

.

Standard deviation

As usual, the standard deviation is obtained from the variance

.

Coefficient of variation

The following results for the coefficient of variation :

.

Crookedness

The skew can be represented as

.

Characteristic function

The characteristic function has the form

with .

Probability generating function

For the probability generating function one obtains

with .

Moment generating function

The moment generating function of the generalized Poisson distribution is

with .