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Math Help - MGF of a random sum

  1. #1
    Member Maccaman's Avatar
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    MGF of a random sum

    Let  X_1, X_2,... ~N(0,1) be an iid sequence and let N ~ Poi (\lambda) independently.

    Find the MGF of the random sum

     S = \sum_{k=1}^N X_k

    What I do know is the pmf and pdf of the N and  X_n , but im confused with the random sum and how to calculate it. Can anyone please help?

    I know that the MGF of the random sum is the product of the MGFs' of the RV's that make up that random sum.
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by Maccaman View Post
    Let  X_1, X_2,... ~N(0,1) be an iid sequence and let N ~ Poi (\lambda) independently.

    Find the MGF of the random sum

     S = \sum_{k=1}^N X_k

    What I do know is the pmf and pdf of the N and  X_n , but im confused with the random sum and how to calculate it. Can anyone please help?

    I know that the MGF of the random sum is the product of the MGFs' of the RV's that make up that random sum.
    1. Find the moment generating function of the sum given N = n (you're accustomed to doing this).

    2. Treat the above result as a function of n and take its expectation with respect to the distribution of N.

    Read this: Compound Poisson distribution - Wikipedia, the free encyclopedia

    Or this (p5): http://www.stats.uwo.ca/faculty/kulp...ndouts/MGF.pdf
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