In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable.
The result can be either a continuous or a discrete distribution.
Suppose that i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and that are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of
random variables is a compound Poisson distribution.
In the case N = 0, then this is a sum of 0 terms, so the value of Y is 0.
Thus Then, since E(N) = Var(N) if N is Poisson-distributed, these formulae can be reduced to The probability distribution of Y can be determined in terms of characteristic functions: and hence, using the probability-generating function of the Poisson distribution, we have An alternative approach is via cumulant generating functions: Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ = 1, the cumulants of Y are the same as the moments of X1.
[citation needed] Every infinitely divisible probability distribution is a limit of compound Poisson distributions.
[1] And compound Poisson distributions is infinitely divisible by the definition.
are positive integer-valued i.i.d random variables with
satisfying probability generating function characterization has a discrete compound Poisson(DCP) distribution with parameters
has a discrete compound Poisson distribution of order
For more special case of DCP, see the reviews paper[7] and references therein.
Feller's characterization of the compound Poisson distribution states that a non-negative integer valued r.v.
[8] The negative binomial distribution is discrete infinitely divisible, i.e., if X has a negative binomial distribution, then for any positive integer n, there exist discrete i.i.d.
random variables X1, ..., Xn whose sum has the same distribution that X has.
This distribution can model batch arrivals (such as in a bulk queue[5][9]).
The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount.
are negative, it is the discrete pseudo compound Poisson distribution.
[3] We define that any discrete random variable
satisfying probability generating function characterization has a discrete pseudo compound Poisson distribution with parameters
The marginal distribution of Y is a Tweedie distribution[10] with variance power 1 < p < 2 (proof via comparison of characteristic function (probability theory)).
To be more explicit, if and i.i.d., then the distribution of is a reproductive exponential dispersion model
is the following: A compound Poisson process with rate
and jump size distribution G is a continuous-time stochastic process
given by where the sum is by convention equal to zero as long as N(t) = 0.
are independent and identically distributed random variables, with distribution function G, which are also independent of
[11] For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models.
[12] A compound Poisson distribution, in which the summands have an exponential distribution, was used by Revfeim 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 has an exponential distribution.
[13] Thompson applied the same model to monthly total rainfalls.
[14] There have been applications to insurance claims[15][16] and x-ray computed tomography.