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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 distribution|continuous]] or a [[discrete distribution]].
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| ==Definition==
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| Suppose that
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| :<math>N\sim\operatorname{Poisson}(\lambda),</math>
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| i.e., ''N'' is a [[random variable]] whose distribution is a [[Poisson distribution]] with [[expected value]] λ, and that
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| :<math>X_1, X_2, X_3, \dots</math>
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| are identically distributed random variables that are mutually independent and also independent of ''N''. Then the probability distribution of the sum of <math>N</math> i.i.d. random variables conditioned on the number of these variables (<math>N</math>):
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| :<math>Y | N=\sum_{n=1}^N X_n</math>
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| 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.
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| 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''.
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| ==Properties==
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| [[Mean]] and [[variance]] of the compound distribution derive in a simple way from [[law of total expectation]] and the [[law of total variance]]. Thus
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| :<math>\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) ,</math>
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| :<math>\operatorname{Var}_Y(Y) = E_N\left[\operatorname{Var}_{Y|N}(Y)\right] + \operatorname{Var}_N\left[E_{Y|N}(Y)\right]
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| =\operatorname{E}_N\left[N\operatorname{Var}_X(X)\right] + \operatorname{Var}_N\left[N\operatorname{E}_X(X)\right] ,</math>
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| giving
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| :<math>\operatorname{Var}_Y(Y) =
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| \operatorname{E}_N(N)\operatorname{Var}_X(X) + \left(\operatorname{E}_X(X)\right)^2\operatorname{Var}_N(N) .</math>
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| Then, since E(''N'')=Var(''N'') if ''N'' is Poisson, and dropping the unnecessary subscripts, these formulae can be reduced to
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| :<math>\operatorname{E}(Y)= \operatorname{E}(N)\operatorname{E}(X) ,</math>
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| :<math>\operatorname{Var}(Y) = E(N)(\operatorname{Var}(X) + {E(X)}^2 )= E(N){E(X^2)}.</math>
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| The probability distribution of ''Y'' can be determined in terms of [[Characteristic function (probability theory)|characteristic function]]s:
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| :<math>\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), \,</math>
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| and hence, using the [[Probability-generating function#Examples|probability generating function]] of the Poisson distribution,
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| :<math>\varphi_Y(t) = \textrm{e}^{\lambda(\varphi_X(t) - 1)}.\,</math>
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| An alternative approach is via [[cumulant generating function]]s:
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| :<math>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)) . \,</math>
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| Via the [[law of total cumulance]] it can be shown that, if the mean of the Poisson distribution λ=1, the [[cumulant]]s of ''Y'' are the same as the [[moment (mathematics)|moments]] of ''X''<sub>1</sub>.{{Citation needed|date=October 2010}}
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| It can be shown that every [[infinite divisibility|infinitely divisible]] probability distribution is a limit of compound Poisson distributions.{{Citation needed|date=October 2010}}
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| ==Special cases==
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| When <math>X_1, X_2, X_3, \dots</math> 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]]).<ref>{{cite jstor|3007241}}</ref><ref>{{cite journal | title = "Stuttering - Poisson" distributions | first = C. D. | last = Kemp | journal = Journal of the Statistical and Social Enquiry of Ireland | year = 1967 | volume = 21 | issue = 5 | pages = 151–157 | url =http://hdl.handle.net/2262/6987}}</ref>
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| 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 distribution]]s. 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.<ref name=Revf>Revfeim, K.J.A. (1984) An initial model of the relationship between rainfall events and daily rainfalls. Journal of Hydrology, 75, 357-364.</ref><ref name=Thom>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.</ref>
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| ==Compound Poisson processes==
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| A [[compound Poisson process]] with rate <math>\lambda>0</math> and jump size distribution ''G'' is a continuous-time [[stochastic process]] <math>\{\,Y(t) : t \geq 0 \,\}</math> given by
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| :<math>Y(t) = \sum_{i=1}^{N(t)} D_i,</math>
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| where the sum is by convention equal to zero as long as ''N''(''t'')=0. Here, <math> \{\,N(t) : t \geq 0\,\}</math> is a [[Poisson process]] with rate <math>\lambda</math>, and <math> \{\,D_i : i \geq 1\,\}</math> are independent and identically distributed random variables, with distribution function ''G'', which are also independent of <math> \{\,N(t) : t \geq 0\,\}.\,</math>{{Citation needed|date=October 2010}}
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| ==Applications==
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| A compound Poisson distribution, in which the summands have an [[exponential distribution]], was used by Revfeim<ref name="Revf"/> 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<ref name="Thom"/> applied the same model to monthly total rainfalls.
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| The discrete compound Poisson distribution is also widely used in [[actuarial science]] <ref>{{Cite journal |first =Zhang | last = Huiming | coauthors = [[Lili Chu]], [[Yu Diao]]|title=Some Properties of the Generalized Stuttering Poisson Distribution and its Applications |journal=[[Studies in Mathematical Sciences]] |volume=5 |issue=1 |year=2012|pages=11–26 |url=http://cscanada.net/index.php/sms/article/view/j.sms.1923845220120501.Z0697 |doi=10.3968/j.sms.1923845220120501.Z0697}}</ref> for modelling the distribution of the total claim amount <math>Y</math> of the sum of a Poisson random number <math>N</math> of independent and identically distributed claim amounts ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub>.
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| {{More footnotes|date=November 2010}}
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| ==References==
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| {{Reflist}}
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| {{ProbDistributions|families}}
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| {{DEFAULTSORT:Compound Poisson Distribution}}
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| [[Category:Probability distributions]]
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| [[Category:Theory of probability distributions]]
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| [[Category:Poisson processes]]
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| [[Category:Compound distributions]]
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It begins by consuming in a pair from the lovely island where that peaceful village is in beaches and woods right up until the enemies known because BlackGuard led by Lieutenant Hammerman invades your island. After managing to guard against a tiny invasion force, he purports to avenge his loss in battle.
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Stay aware of how multi player works. In case that you're investing in a game exclusively for it has the multiplayer, be sure the person have everything required intended for this. If you'll be planning on playing against a person in your prized household, you may stumble on that you will want two copies of specific clash of clans cheats to play against one another.
Assuming that you're playing a gameplay online, and you launch across another player of which seems to be discouraging other players (or you, in particular) intentionally, really don't take it personally. This is called "Griefing," and it's the playing games equivalent of Internet trolling. Griefers are just out for negative attention, and you give people what they're looking with regard to if you interact together. Don't get emotionally put in in what's happening as simply try to overlook it.
Provide you with the in-online game songs the opportunity. If, nonetheless, you might end annoyed by using who's soon after one per hour approximately, don't be worried to mute the television set or personal computer and as a consequence play some audio of one's very own. You will discover a far more satisfying game playing experience performing this and therefore are way more unlikely to get the new frustration from actively understanding.
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