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EXAMPLE: SupposeX1,X2, ...,Xnare independentPoissonrandom variables and define the sumYX1X2...Xn. It followsfrom previous properties of expected value and variance thatEYnandVarYn. Because the variance equals the mean, it is temptingto concludeYhas theYPoissonndistributions. But we need toshow more than that the first two moments ofYmatch with thePoissonndistribution.103
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∙To showYPoissonnwe use the MGF approach withitexpexpt−1for eachi:Yti1nexpexpt−1exp∑i1nexpt−1expnexpt−1and the final expression is the MGF of thePoissonndistribution.∙We can extend this result to allow different means: ifXihas thePoissonidistribution then the sum has thePoisson12...ndistribution.104