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Thread: Reversing the moment generating function

  1. #1
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    Reversing the moment generating function

    So this is my moment generating function


    Now i need to find the PDF or CDF

    So i think the moment genereting function needs to be divided by e^{tx}

    But how do i do this in an algabraic way
    This is what i got so far


    And now i'm stuck

  2. #2
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    Re: Reversing the moment generating function

    Note that the mgf of a random variable X is defined by

    M_X(t) = E[e^{tX}]

    and mathematically speaking it is a Laplace transform of the corresponding pmf/pdf. Now you are given the mgf, and want to find the corresponding pmf/pdf, so generally speaking you would like to do an inverse-Laplace transform. There are table for this for many common functional form.

    However, your mgf is easy enough to recognize. Think about how do you calculate the mgf for a discrete distribution, say a Binomial distribution. Then you can match the corresponding probabilities and support points. Also note that the Laplace transform is unique so you will have a unique solution.

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