Inprobability theory, acompound Poisson distribution is theprobability distribution of the sum of a number ofindependent identically-distributed random variables, where the number of terms to be added is itself aPoisson-distributed variable. The result can be either acontinuous or adiscrete distribution.

Suppose that

${\displaystyle N\sim \mathrm{Poisson}(\lambda ),}$

i.e.,N is arandom variable whose distribution is aPoisson distribution withexpected value λ, and that

${\displaystyle X_1,X_2,X_3,\dots }$

are identically distributed random variables that are mutually independent and also independent ofN. Then the probability distribution of the sum of${\displaystyle N}$ i.i.d. random variables

${\displaystyle Y=\sum _{n=1}^N{X}_n}$

is a compound Poisson distribution.

In the caseN = 0, then this is a sum of 0 terms, so the value ofY is 0. Hence the conditional distribution ofY given thatN = 0 is adegenerate distribution.

The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) overN, and this joint distribution can be obtained by combining the conditional distributionY | N with the marginal distribution ofN.

Theexpected value and thevariance of the compound distribution can be derived in a simple way fromlaw of total expectation and thelaw of total variance. Thus

${\displaystyle \mathrm{E}(Y)=\mathrm{E}\left[\mathrm{E}(Y\mid N)\right]=\mathrm{E}\left[N\mathrm{E}(X)\right]=\mathrm{E}(N)\mathrm{E}(X),}$

Then, since E(N) = Var(N) ifN is Poisson-distributed, these formulae can be reduced to

${\displaystyle \mathrm{E}(Y)=\mathrm{E}(N)\mathrm{E}(X)=\lambda \mathrm{E}(X),}$

${\displaystyle \mathrm{Var}(Y)=\mathrm{E}(N)(\mathrm{Var}(X)+(\mathrm{E}(X){)}^2)=\mathrm{E}(N)\mathrm{E}(X^2)=\lambda \mathrm{E}(X^2).}$

The probability distribution ofY can be determined in terms ofcharacteristic functions:

${\displaystyle {\phi }_Y(t)=\mathrm{E}(e^{itY})=\mathrm{E}\left({\left(\mathrm{E}(e^{itX}\mid N)\right)}^N\right)=\mathrm{E}\left(({\phi }_X(t){)}^N\right),}$

and hence, using theprobability-generating function of the Poisson distribution, we have

${\displaystyle {\phi }_Y(t)={\text{e}}^{\lambda ({\phi }_X(t)-1)}.}$

An alternative approach is viacumulant generating functions:

${\displaystyle K_Y(t)=\ln \mathrm{E}[e^{tY}]=\ln \mathrm{E}[\mathrm{E}[e^{tY}\mid N]]=\ln \mathrm{E}[e^{N{K}_X(t)}]=K_N(K_X(t)).}$

Via thelaw of total cumulance it can be shown that, if the mean of the Poisson distributionλ = 1, thecumulants ofY are the same as themoments ofX₁.^{[citation needed]}

Everyinfinitely divisible probability distribution is a limit of compound Poisson distributions.^[1] And compound Poisson distributions is infinitely divisible by the definition.

When${\displaystyle X_1,X_2,X_3,\dots }$ are positive integer-valued i.i.d random variables with${\displaystyle P(X_1=k)={\alpha }_k,(k=1,2,\dots )}$, then this compound Poisson distribution is nameddiscrete compound Poisson distribution^[2][3][4] (or stuttering-Poisson distribution^[5]) . We say that the discrete random variable${\displaystyle Y}$ satisfyingprobability generating function characterization

${\displaystyle P_Y(z)=\sum _{i=0}^{\mathrm{\infty }}P(Y=i)z^i=\exp \left(\sum _{k=1}^{\mathrm{\infty }}{\alpha }_k\lambda (z^k-1)\right),(|z|\le 1)}$

has a discrete compound Poisson(DCP) distribution with parameters${\displaystyle ({\alpha }_1\lambda ,{\alpha }_2\lambda ,\dots )\in {\mathbb{R}}^{\mathrm{\infty }}}$ (where$\sum _{i=1}^{\mathrm{\infty }}{\alpha }_i=1$, with${\alpha }_i\ge 0,\lambda >0$), which is denoted by

${\displaystyle X\sim \text{DCP}(\lambda {\alpha }_1,\lambda {\alpha }_2,\dots )}$

Moreover, if${\displaystyle X\sim \mathrm{DCP}(\lambda {\alpha }_1,\dots ,\lambda {\alpha }_r)}$, we say${\displaystyle X}$ has a discrete compound Poisson distribution of order${\displaystyle r}$ . When${\displaystyle r=1,2}$, DCP becomesPoisson distribution andHermite distribution, respectively. When${\displaystyle r=3,4}$, DCP becomes triple stuttering-Poisson distribution and quadruple stuttering-Poisson distribution, respectively.^[6] Other special cases include: shiftgeometric distribution,negative binomial distribution,Geometric Poisson distribution,Neyman type A distribution, Luria–Delbrück distribution inLuria–Delbrück experiment. 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.${\displaystyle X}$ isinfinitely divisible if and only if its distribution is a discrete compound Poisson distribution.^[8] Thenegative binomial distribution is discreteinfinitely divisible, i.e., ifX has a negative binomial distribution, then for any positive integern, there exist discrete i.i.d. random variablesX₁, ..., X_n whose sum has the same distribution thatX has. The shiftgeometric distribution is discrete compound Poisson distribution since it is a trivial case ofnegative binomial distribution.

This distribution can model batch arrivals (such as in abulk queue^[5][9]). The discrete compound Poisson distribution is also widely used inactuarial science for modelling the distribution of the total claim amount.^[3]

When some${\displaystyle {\alpha }_k}$ are negative, it is the discrete pseudo compound Poisson distribution.^[3] We define that any discrete random variable${\displaystyle Y}$ satisfyingprobability generating function characterization

${\displaystyle G_Y(z)=\sum _{i=0}^{\mathrm{\infty }}P(Y=i)z^i=\exp \left(\sum _{k=1}^{\mathrm{\infty }}{\alpha }_k\lambda (z^k-1)\right),(|z|\le 1)}$

has a discrete pseudo compound Poisson distribution with parameters${\displaystyle ({\lambda }_1,{\lambda }_2,\dots )=:({\alpha }_1\lambda ,{\alpha }_2\lambda ,\dots )\in {\mathbb{R}}^{\mathrm{\infty }}}$ where$\sum _{i=1}^{\mathrm{\infty }}{\alpha }_i=1$ and, with${\displaystyle {\alpha }_i\in \mathbb{R},\lambda >0}$.

IfX has agamma distribution, of which theexponential distribution is a special case, then the conditional distribution ofY | N is again a gamma distribution. The marginal distribution ofY is aTweedie distribution with variance power 1 < p < 2 (proof via comparison ofcharacteristic function).^[10] To be more explicit, if

${\displaystyle N\sim \mathrm{Poisson}(\lambda ),}$

and

${\displaystyle X_i\sim \mathrm{\Gamma }(\alpha ,\beta )}$

i.i.d., then the distribution of

${\displaystyle Y=\sum _{i=1}^N{X}_i}$

is a reproductiveexponential dispersion model${\displaystyle ED(\mu ,{\sigma }^2)}$ with

The mapping of parameters Tweedie parameter${\displaystyle \mu ,{\sigma }^2,p}$ to the Poisson and Gamma parameters${\displaystyle \lambda ,\alpha ,\beta }$ is the following:

Acompound Poisson process with rate${\displaystyle \lambda >0}$ and jump size distributionG is a continuous-timestochastic process${\displaystyle \{Y(t):t\ge 0\}}$ given by

${\displaystyle Y(t)=\sum _{i=1}^{N(t)}{D}_i,}$

where the sum is by convention equal to zero as long asN(t) = 0. Here,${\displaystyle \{N(t):t\ge 0\}}$ is aPoisson process with rate${\displaystyle \lambda }$, and${\displaystyle \{D_i:i\ge 1\}}$ are independent and identically distributed random variables, with distribution functionG, which are also independent of${\displaystyle \{N(t):t\ge 0\}.}$^[11]

For the discrete version of compound Poisson process, it can be used insurvival analysis for the frailty models.^[12]

A compound Poisson distribution, in which the summands have anexponential 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 toinsurance claims^[15][16] andx-ray computed tomography.^[17][18][19]

• Discrete distributions
• Poisson distribution
• Infinitely divisible probability distributions
• Compound probability distributions

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