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Math Help - Power Series Distribution - MGF and PGF

  1. #1
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    Power Series Distribution - MGF and PGF

    I'm having some trouble with the following problem, and to be frank, I'm not sure where to begin.

    For the discrete random variable with probability mass function

    p_{X}(x)=\frac{a(x)\theta^x}{c(\theta)}I[x\in\{0,1,2,3...\}]

    a(x)\geq(0); c(\theta)>0

    Find the moment generating function and the probability generating function.

    I have a feeling I need to write out the terms and use properties of derivatives as they relate to convergent series to get this answer, but I'm not entirely sure.

    Some pointers in the right direction would be much appreciated. Thank you.
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  2. #2
    MHF Contributor matheagle's Avatar
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    The a(x) makes this a mess.

    Since E(e^{tX})={1\over c(\theta)}\sum_{x=0}^{\infty}a(x) \left(e^t\theta\right)^x
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