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Example suppose x 1 x 2 x n are independent poisson

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EXAMPLE : Suppose X 1 , X 2 , ..., X n are independent Poisson random variables and define the sum Y X 1 X 2 ... X n . It follows from previous properties of expected value and variance that E Y n and Var Y n . Because the variance equals the mean, it is tempting to conclude Y has the Y Poisson n distributions. But we need to show more than that the first two moments of Y match with the Poisson n distribution. 103
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To show Y Poisson n we use the MGF approach with i t exp exp t 1  for each i : Y t i 1 n exp exp t 1  exp i 1 n exp t 1 exp n exp t 1  and the final expression is the MGF of the Poisson n distribution. We can extend this result to allow different means: if X i has the Poisson i distribution then the sum has the Poisson 1 2 ... n distribution. 104
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