1. 12. Compound Poisson Process. Let c05-math-480 be a Poisson process with intensity c05-math-481 defined on the probability space c05-math-482 with respect to the filtration c05-math-483, and let c05-math-484 be a sequence of independent and identically distributed random variables with common mean c05-math-485 and variance c05-math-486. Let c05-math-487 be independent of c05-math-488. By defining the compound Poisson process c05-math-489 as
    equation

    show that the moment generating function for is

    where .

    Further, show that ...

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