Mixed Poisson distribution
The mixed Poisson distribution is a probability distribution in stochastics that is univariate and one of the discrete probability distributions . It can be found as a general approach for the distribution of loss numbers in actuarial mathematics . It generalizes the Poisson distribution and should not be confused with the composite Poisson distribution .
definition
A random variable is sufficient for the mixed Poisson distribution with the density if it is the probabilities
owns. If we denote the probabilities of the Poisson distribution with , then we have
- .
properties
- The variance is always greater than the expected value . This property is called About dispersion ( English over-dispersion ). This is in contrast to the Poisson distribution, where the expectation and variance are identical.
- In practice only densities of gamma distributions , logarithmic normal distributions and of inverse Gaussian distributions are used as densities . If you choose the density of the gamma distribution, you get the negative binomial distribution , which explains why this is also called the Poisson gamma distribution.
The following is the expected value of the density and the variance of this density.
Expected value
The expected value results in
- .
Variance
For the variance one gets
- .
Standard deviation
The standard deviation is obtained from the expected value and 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
- .
It is the moment generating function of the density.
Probability generating function
For the probability generating function one obtains
- .
Moment generating function
The moment generating function of the mixed Poisson distribution is
- .
literature
- Jan Grandell: Mixed Poisson Processes. Chapman & Hall, London 1997, ISBN 0-412-78700-8 .