Composite Poisson distribution

The composite Poisson distribution is a generalization of the Poisson distribution and plays an important role in Poisson processes and the theory of infinite divisibility . In contrast to many other distributions, the composite Poisson distribution does not determine a priori whether it is continuous or discrete. It should not be confused with the mixed Poisson distribution .

definition

If a Poisson-distributed random variable with expected value and if there are independently and identically distributed random variables , then the random variable is called ${\displaystyle N}$ ${\displaystyle \lambda }$ ${\displaystyle (X_{i})_{i\in \mathbb {N} }}$

${\displaystyle Y:=\sum _{i=1}^{N}X_{i}}$

composed Poisson-distributed . If they are all defined, i.e. discrete, then discrete compound is called Poisson distributed . In both cases one writes where is the probability measure of . Probability densities or probability functions as well as distribution functions can only be specified in closed form in special cases, but can possibly be approximated with the Panjer algorithm . ${\displaystyle X_{i}}$ ${\displaystyle \mathbb {N} _{0}}$ ${\displaystyle Y}$ ${\displaystyle Y\sim CPoi_{\mu }}$ ${\displaystyle \mu }$ ${\displaystyle X_{i}}$

Occasionally, the terms compound Poisson and discrete compound Poisson are used in German literature .

properties

Expected value

According to Wald's formula, the following applies to the expected value :

${\displaystyle \operatorname {E} (Y)=\lambda \operatorname {E} (X_{1})}$.

Variance

According to the Blackwell-Girshick equation ,

${\displaystyle \operatorname {Var} (Y)=\lambda \,(\operatorname {E} (X_{1}))^{2}+\lambda \operatorname {Var} (X_{1})=\lambda \operatorname {E} (X_{1}^{2}),}$

when the second moments of exist. The second equality follows from the law of displacement . ${\displaystyle X_{i}}$

Crookedness

Using the cumulative results for the skewness

${\displaystyle \operatorname {v} (Y)={\frac {\lambda \operatorname {E} (X_{1}^{3})}{\left(\lambda \operatorname {E} (X_{1}^{2})\right)^{\frac {3}{2}}}}}$.

Bulge

For the excess , the cumulative results

${\displaystyle \gamma ={\frac {\operatorname {E} (X_{1}^{4})}{\lambda \operatorname {E} (X_{1}^{2})^{2}}}}$.

Accumulators

The cumulative generating function is

${\displaystyle g_{X}(t)=\lambda (M_{X_{1}}(t)-1)}$

where is the moment generating function of . This applies to all accumulators ${\displaystyle M_{X_{1}}(t)}$ ${\displaystyle X_{1}}$

${\displaystyle \kappa _{k}=\lambda \operatorname {E} (X_{1}^{k})}$.

Moment generating function

The moment-generating function results from the concatenation of the probability-generating function of the Poisson distribution and the moment-generating function of : ${\displaystyle X_{i}}$

${\displaystyle M_{Y}(t)=\exp(\lambda (M_{X_{1}}(t)-1))}$.

Characteristic function

The characteristic function results from the concatenation of the probability-generating function of the Poisson distribution and the characteristic function of : ${\displaystyle X_{i}}$

${\displaystyle \varphi _{Y}(t)=\exp(\lambda (\varphi _{X_{1}}(t)-1))}$

Probability generating function

If they are discrete, then the probability-generating function is defined and results from a concatenation of the probability-generating function from and from to ${\displaystyle X_{i}}$ ${\displaystyle N}$ ${\displaystyle X_{i}}$

${\displaystyle m_{Y}(t)=\exp(\lambda (m_{X_{1}}(t)-1))}$.

Infinite divisibility

A composite Poisson random variable is infinitely divisible . It can be shown that a random variable is infinitely divisible if and only if the random variable is composed of discrete Poisson distribution. ${\displaystyle \mathbb {N} _{0}}$

Relationship to other distributions

Relationship to the Poisson distribution

Is almost certain , so fall Poisson distribution and compound Poisson distribution together. ${\displaystyle X_{i}=1}$

Relationship to the geometric distribution and to the negative binomial distribution

Since both the geometric distribution and the negative binomial distribution are infinitely divisible, they are composite Poisson distributions. They arise when combined with the logarithmic distribution . The parameters of the negative binomial distribution are calculated as and . ${\displaystyle p_{\text{log}}=1-p_{\text{neg}}}$ ${\displaystyle r={\frac {-\lambda }{\ln(1-p_{\text{log}})}}}$

Web links

• AV Prokhorov: Poisson distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .