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ⓘ Compound Poisson distribution. In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent iden ..




                                     

ⓘ Compound Poisson distribution

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. In the simplest cases, the result can be either a continuous or a discrete distribution.

                                     

1. Definition

Suppose that

N ∼ Poisson ⁡ λ, {\displaystyle N\sim \operatorname {Poisson} \lambda,}

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

X 1, X 2, X 3, … {\displaystyle 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 of N {\displaystyle N} i.i.d. random variables

Y = ∑ n = 1 N X n {\displaystyle Y=\sum _{n=1}^{N}X_{n}}

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. Hence the conditional distribution of Y given that N = 0 is a degenerate distribution.

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

                                     

2. Properties

The expected value and the variance of the compound distribution can be derived in a simple way from law of total expectation and the law of total variance. Thus

E ⁡ Y = E ⁡ =K_{N}K_{X}t).\,}

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 X 1.

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

                                     

3. Discrete compound Poisson distribution

When X 1, X 2, X 3, … {\displaystyle X_{1},X_{2},X_{3},\dots } are non-negative integer-valued i.i.d random variables with P X 1 = k = α k, k = 1, 2, … {\displaystyle PX_{1}=k=\alpha _{k},\ k=1.2,\ldots}, then this compound Poisson distribution is named discrete compound Poisson distribution or stuttering-Poisson distribution. We say that the discrete random variable Y {\displaystyle Y} satisfying probability generating function characterization

P Y z = ∑ i = 0 ∞ P Y = i z i = exp ⁡ ∑ k = 1 ∞ α k λ z k − 1), | z | ≤ 1 {\displaystyle P_{Y}z=\sum \limits _{i=0}^{\infty }PY=iz^{i}=\exp \left\sum \limits _{k=1}^{\infty }\alpha _{k}\lambda z^{k}-1\right),\quad |z|\leq 1}

has a discrete compound PoissonDCP distribution with parameters ∈ R ∞ {\displaystyle \alpha _{1}\lambda,\alpha _{2}\lambda,\ldots\in \mathbb {R} ^{\infty }\left\sum _{i=1}^{\infty }\alpha _{i}=1,\alpha _{i}\geq 0,\lambda > 0\right}, which is denoted by

X ∼ DCP {\displaystyle X\sim {\text{DCP}}\lambda {\alpha _{1}},\lambda {\alpha _{r}},\ldots}

Moreover, if X ∼ DCP λ α 1, …, λ α r {\displaystyle X\sim {\operatorname {DCP} }\lambda {\alpha _{1}},\ldots,\lambda {\alpha _{r}}}, we say X {\displaystyle X} has a discrete compound Poisson distribution of order r {\displaystyle r}. When r = 1, 2 {\displaystyle r=1.2}, DCP becomes Poisson distribution and Hermite distribution, respectively. When r = 3, 4 {\displaystyle r=3.4}, DCP becomes triple stuttering-Poisson distribution and quadruple stuttering-Poisson distribution, respectively. Other special cases include: shiftgeometric distribution, negative binomial distribution, Geometric Poisson distribution, Neyman type A distribution, Luria–Delbruck distribution in Luria–Delbruck experiment. For more special case of DCP, see the reviews paper and references therein.

Fellers characterization of the compound Poisson distribution states that a non-negative integer valued r.v. X {\displaystyle X} is infinitely divisible if and only if its distribution is a discrete compound Poisson distribution. It can be shown that 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 X 1., X n whose sum has the same distribution that X has. The shift geometric distribution is discrete compound Poisson distribution since it is a trivial case of negative binomial distribution.

This distribution can model batch arrivals such as in a bulk queue. The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount.

When some α k {\displaystyle \alpha _{k}} are non-negative, it is the discrete pseudo compound Poisson distribution. We define that any discrete random variable Y {\displaystyle Y} satisfying probability generating function characterization

G Y z = ∑ n = 0 ∞ P Y = n z n = exp ⁡ ∑ k = 1 ∞ α k λ z k − 1), | z | ≤ 1 {\displaystyle G_{Y}z=\sum \limits _{n=0}^{\infty }PY=nz^{n}=\exp \left\sum \limits _{k=1}^{\infty }\alpha _{k}\lambda z^{k}-1\right),\quad |z|\leq 1}

has a discrete pseudo compound Poisson distribution with parameters =: ∈ R ∞ {\displaystyle \lambda _{1},\lambda _{2},\ldots=:\alpha _{1}\lambda,\alpha _{2}\lambda,\ldots\in \mathbb {R} ^{\infty }\left{\sum \limits _{k=1}^{\infty }{\alpha _{k}}=1,\sum \limits _{k=1}^{\infty }{\left|{\alpha _{k}}\right|} 0}\right}.



                                     

4. Applications

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. Thompson applied the same model to monthly total rainfalls.

                                     
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