Compound Poisson distribution

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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.

Definition

Suppose that

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

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

$X_{1},X_{2},X_{3},\dots$

are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of $N$ i.i.d. random variables conditioned on the number of these variables ( $N$ ):

$Y|N=\sum _{{n=1}}^{N}X_{n}$

is a well-defined distribution. In the case N = 0, then the value of Y is 0, so that then Y | N=0 has a degenerate distribution.

The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) over N, where this joint distribution is obtained by combining the conditional distribution Y | N with the marginal distribution of N.

Properties

Mean and variance of the compound distribution derive in a simple way from law of total expectation and the law of total variance. Thus

$\operatorname {E}_{Y}(Y)=\operatorname {E}_{N}\left[\operatorname {E}_{{Y|N}}(Y)\right]=\operatorname {E}_{N}\left[N\operatorname {E}_{X}(X)\right]=\operatorname {E}_{N}(N)\operatorname {E}_{X}(X),$

$\operatorname {Var}_{Y}(Y)=E_{N}\left[\operatorname {Var}_{{Y|N}}(Y)\right]+\operatorname {Var}_{N}\left[E_{{Y|N}}(Y)\right]=\operatorname {E}_{N}\left[N\operatorname {Var}_{X}(X)\right]+\operatorname {Var}_{N}\left[N\operatorname {E}_{X}(X)\right],$

giving

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

Then, since E(N)=Var(N) if N is Poisson, and dropping the unnecessary subscripts, these formulae can be reduced to

$\operatorname {E}(Y)=\operatorname {E}(N)\operatorname {E}(X),$

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

The probability distribution of Y can be determined 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 the mean of the Poisson distribution λ=1, the cumulants of Y are the same as the moments of X₁.^{[citation needed]}

It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions.^{[citation needed]}

Special cases

When $X_{1},X_{2},X_{3},\dots$ are non-negative discrete random variables, then this compound Poisson distribution is named stuttering Poisson distribution (non-negative discrete compound Poisson distribution) which can model batch arrivals (such as in a bulk queue).^[1]^[2] If the distribution of X is either an exponential distribution or a gamma distribution, then the conditional distributions of Y | N are gamma distributions in which the shape parameters are proportional to N. This shows that the formulation of the "compound Poisson distribution" outlined above is essentially the same as the more general class of compound probability distributions. However, the properties outlined above do depend on its formulation as the sum of a Poisson-distributed number of random variables. The distribution of Y in the case of the compound Poisson distribution with exponentially-distributed summands can be written in an form.^[3]^[4]

Compound Poisson processes

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

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

where the sum is by convention equal to zero as long as N(t)=0. Here, $\{\,N(t):t\geq 0\,\}$ is a Poisson process with rate $\lambda$ , and $\{\,D_{i}:i\geq 1\,\}$ are independent and identically distributed random variables, with distribution function G, which are also independent of $\{\,N(t):t\geq 0\,\}.\,$ ^{[citation needed]}

Applications

A compound Poisson distribution, in which the summands have an exponential distribution, was used by Revfeim^[3] 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^[4] applied the same model to monthly total rainfalls.

The discrete compound Poisson distribution is also widely used in actuarial science ^[5] for modelling the distribution of the total claim amount $Y$ of the sum of a Poisson random number $N$ of independent and identically distributed claim amounts X₁, X₂, ..., X_N.

References

↑ Adelson, R. M. (1966). "Compound Poisson Distributions". OR (Operational Research Society) 17 (1): 73–75. JSTOR 3007241.
↑ Kemp, C. D. (1967). ""Stuttering - Poisson" distributions". Journal of the Statistical and Social Enquiry of Ireland 21 (5): 151–157.
↑ 3.0 3.1 Revfeim, K.J.A. (1984) An initial model of the relationship between rainfall events and daily rainfalls. Journal of Hydrology, 75, 357-364.
↑ 4.0 4.1 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.
↑ Huiming, Zhang; Lili Chu, Yu Diao (2012). "Some Properties of the Generalized Stuttering Poisson Distribution and its Applications". Studies in Mathematical Sciences 5 (1): 11–26. doi:10.3968/j.sms.1923845220120501.Z0697. Cite uses deprecated parameters (help)

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
Erlang
exponential
F
folded normal
Flory-Schulz
Fréchet
Gamma
Gamma/Gompertz
generalized inverse Gaussian
Gompertz
half-logistic
half-normal
Hotelling's T-squared
hyper-Erlang
hyperexponential
hypoexponential
inverse chi-squared (scaled inverse chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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