Cox process

The mixed Poisson distribution is a discrete probability distribution, which is found as a general approach for the damage index distribution in insurance mathematics. It generalizes the Poisson distribution.

Definition

A random variable satisfies the mixed Poisson distribution with the density when the probabilities

Possesses. If we denote the probabilities of the Poisson distribution, therefore applies

Properties

  • The variance is always greater than the expected value. This property is called overdispersion ( engl. Over dispersion). This is in contrast to the Poisson distribution, are the same for the expectation value and variance.
  • In practice only be used as densities densities of gamma distributions, log- normal distributions and of the inverse Gaussian distributions.
  • Is the density of a gamma distribution, the Poisson distribution is mixed a negative binomial distribution.

In the following, the expected value of the density, and the variance of the density.

Expected value

The expected value is given by

Variance

For the variance we obtain

Standard deviation

From expectation and variance gives the standard deviation

Coefficient of variation

For the coefficient of variation results in:

Skew

The skewness can be represented as

Characteristic function

The characteristic feature is in the form

Generating function

For the generating function is obtained

Moment generating function

The moment generating function of the mixed Poisson distribution is

  • Probability distribution
Actuarial science Log-normal distribution Characteristic function (probability theory)
365283
de