Compound Poisson distribution

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In probability theory, a compound Poisson distribution is the probability distribution of the sum of a "Poisson-distributed number" of independent identically-distributed random variables. More precisely, suppose

N\sim\operatorname{Poisson}(\lambda),

i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and

X_1, X_2, X_3, \dots

are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum

Y=\sum_{n=1}^N X_n

is a compound Poisson distribution. (When N = 0, then the value of Y is 0.)

Contents

Properties

In terms of the basic moments,

E(Y) = E(X) E(N)\,
\operatorname{Var}(Y) = E(N)\operatorname{Var}(X) + {E(X)}^2 \operatorname{Var}(N) ,

and, since E(N)=Var(N) if N is Poisson, this can be reduced to

\operatorname{Var}(Y) = E(N)(\operatorname{Var}(X) + {E(X)}^2 ) .


In terms of characteristic functions,

\varphi_Y(t) = \operatorname{E}\left(e^{itY}\right)= \operatorname{E}_N\left( \left(\operatorname{E}\left(e^{itX}\right) \right)^{N} \right)= \operatorname{E}_N\left( \left(\varphi_X(t) \right)^{N} \right), \,

and hence, using the probability generating function of the Poisson distribution,

\varphi_Y(t) = \textrm{e}^{\lambda(\varphi_X(t) - 1)}.\,

An alternative approach is via cumulant generating functions:

K_Y(t)=\mbox{ln} E[e^{tY}]=\mbox{ln} E[E[e^{tY}|N]]=\mbox{ln} E[e^{NK_X(t)}]=K_N(K_X(t)) . \,


Via the law of total cumulance it can be shown that, if λ=1, the moments of X1 are the cumulants of Y.

It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions.

Compound Poisson processes

A compound Poisson process with rate λ > 0 and jump size distribution G is a continuous-time stochastic process \{\,Y(t) : t \geq 0 \,\} given by

Y(t) = \sum_{i=0}^{N(t)} D_i

where, \{\,N(t) : t \geq 0\,\} is a Poisson process with rate λ, and \{\,D_i : i \geq 0\,\} are independent and identically distributed random variables, with distribution function G, which are also independent of \{\,N(t) : t \geq 0\,\}.\,

Applications

A compound Poisson distribution, in which the summands have an exponential distribution, was used by Revfeim[1] to model the distribution of the total rainfall in a day, where each day contains a Poisson-distributed number of events each of which provides an amount of rainfall which is has an exponential distribution. Thompson[2] applied the same model to monthly total rainfalls.

References

  1. ^ Revfeim, K.J.A. (1984) An initial model of the relationship between rainfall events and daily rainfalls. Journal of Hydrology, 75, 357-364.
  2. ^ Thompson, C.S. (1984) Homogeneity analysis of a rainfall series: an application of the use of a realistic rainfall model. J. Climatology, 4, 609 – 619.
Probability distributions
 
Discrete univariate with infinite support
Retrieved from "http://en.wikipedia.org/wiki/Compound_Poisson_distribution"

This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.

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