Compound Poisson distribution: Information from Answers.com

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

1 Properties
2 Compound Poisson processes
3 Applications
4 References

Properties

In terms of the basic moments,

$E(Y) = E(X) E(N)\,$

(Proof: By the law of total expectation we have:

$E_Y[Y]= E_N\left[E_{Y|N}[Y]\right]= E_N\left[NE_X[X]\right]=E_N[N]E_X[X]$

)

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

(Proof: By the law of total variance we have:

$\operatorname{Var}_Y[Y] = E_N\left[\operatorname{Var}_{Y|N}[Y]\right] + \operatorname{Var}_N\left[E_{Y|N}[Y]\right] \Leftrightarrow$

$\operatorname{Var}_Y[Y]= E_N\left[N\operatorname{Var}_X[X]\right] + \operatorname{Var}_N\left[NE_X[X]\right] = E_N[N]\operatorname{Var}_X[X] + \left(E_X[X]\right)^2\operatorname{Var}_N[N]$

)

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

^ Revfeim, K.J.A. (1984) An initial model of the relationship between rainfall events and daily rainfalls. Journal of Hydrology, 75, 357-364.
^ 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 finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · 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]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · 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 · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

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Compound Poisson distribution

Contents

Properties

Compound Poisson processes

Applications

References