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Math Help - Finding probability function of moment generating function

  1. #1
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    Finding probability function of moment generating function

    Hi,

    I have the following moment generating function:
    M_X(t) = 0.2e^{-3t}+0.11+0.1e^t+0.34e^{2t}+0.05e^{3t}+0.2e^{10t}

    I would like to find the support and the value of f_X(0) (i.e. the probability function of X evaluated at 0), but I am unsure how to go about doing this. I'm just guessing here, but couldn't X take on any value?

    I also know that a moment generating function is defined as
    M_X(t) = \sum_{x \in X}e^{tx}f(x)
    So I factored e^t out of the original equation, but I'm not sure how to get the original equation into a summation in terms of x. Can someone help me understand this better?
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  2. #2
    Guy
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    Re: Finding probability function of moment generating function

    This seems to take advantage of fact that e^{xt} are linearly independent. In order to get the term .2 e^{-3t}, what must happen? In particular, does this imply that P(X = x) = .2 for some x? Similar logic applies to each term in the summation.
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  3. #3
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    Re: Finding probability function of moment generating function

    I'm sorry, but I'm not quite sure I understand. I have a feeling that x=-3 would produce 0.2, but I can't justify it.
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  4. #4
    Guy
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    Re: Finding probability function of moment generating function

    Good. Now, do that for each term and use that to conjure up a pmf (probability mass function). Then, check that the pmf does, indeed, have that mgf.
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  5. #5
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    Re: Finding probability function of moment generating function

    Ahh, that makes sense. So does that mean X can only take on the values -3,0,1,2,3,10?
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  6. #6
    Guy
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    Re: Finding probability function of moment generating function

    Effectively, yes. It means that, if X has the distribution associated with that pmf then X takes on the values you mentioned with probability 1.
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  7. #7
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    Re: Finding probability function of moment generating function

    Perfect. Thank you so much!
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