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 ${\displaystyle X}$ ${\displaystyle \pi (\lambda )}$

${\displaystyle \operatorname {P} (X=k)=p_{\pi }(k)=\int \limits _{0}^{\infty }{\frac {\lambda ^{k}}{k!}}e^{-\lambda }\,\,\pi (\lambda )\,\mathrm {d} \lambda }$

owns. If we denote the probabilities of the Poisson distribution with , then we have ${\displaystyle q_{\lambda }(k)\,}$

${\displaystyle \operatorname {P} (X=k)=p_{\pi }(k)=\int \limits _{0}^{\infty }q_{\lambda }(k)\,\,\pi (\lambda )\,\mathrm {d} \lambda }$.

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. ${\displaystyle \pi (\lambda )}$

The following is the expected value of the density and the variance of this density. ${\displaystyle \mu _{\pi }=\int \limits _{0}^{\infty }\lambda \,\,\pi (\lambda )d\lambda \,}$ ${\displaystyle \pi (\lambda )\,}$ ${\displaystyle \sigma _{\pi }^{2}=\int \limits _{0}^{\infty }(\lambda -\mu _{\pi })^{2}\,\,\pi (\lambda )d\lambda \,}$

Expected value

The expected value results in

${\displaystyle \operatorname {E} (X)=\mu _{\pi }}$.

Variance

For the variance one gets

${\displaystyle \operatorname {Var} (X)=\mu _{\pi }+\sigma _{\pi }^{2}}$.

Standard deviation

The standard deviation is obtained from the expected value and the variance

${\displaystyle \sigma ={\sqrt {\mu _{\pi }+\sigma _{\pi }^{2}}}}$.

Coefficient of variation

The following results for the coefficient of variation :

${\displaystyle \operatorname {VarK} (X)={\sqrt {\frac {\mu _{\pi }+\sigma _{\pi }^{2}}{\mu _{\pi }^{2}}}}}$.

Crookedness

The skew can be represented as

${\displaystyle \operatorname {v} (X)={\Bigl (}\mu _{\pi }+\sigma _{\pi }^{2}{\Bigr )}^{-{\frac {3}{2}}}\,{\Biggl [}\int \limits _{0}^{\infty }(\lambda -\mu _{\pi })^{3}\,\pi (\lambda )\,d{\lambda }+\mu _{\pi }{\Biggr ]}}$.

Characteristic function

The characteristic function has the form

${\displaystyle \varphi _{X}(s)=M_{\pi }(e^{is}-1)\,}$.

It is the moment generating function of the density. ${\displaystyle M_{\pi }}$

Probability generating function

For the probability generating function one obtains

${\displaystyle m_{X}(s)=M_{\pi }(s-1)\,}$.

Moment generating function

The moment generating function of the mixed Poisson distribution is

${\displaystyle M_{X}(s)=M_{\pi }(e^{s}-1)\,}$.

literature

• Jan Grandell: Mixed Poisson Processes. Chapman & Hall, London 1997, ISBN 0-412-78700-8 .