Q's 1)How to find PDF from MGF?; 2)A problem

Discussion in 'Probability' started by Rueful Rabbit, Nov 20, 2011.

  1. Hi All,

    I have an MGF and I want to find its PDF.
    The advice at http://mathforum.org/library/drmath/view/74480.html
    is ...
    f(x) = 1/(2pi) * integral from -infinity to infinity of M_Y(it) * exp(-
    itx) dt

    But I could not find any other reference. Know or any good links?

    But I hear you ask, "Why dear fool would you want such a thing?"

    Well, I'm given the following MGF ...
    M_w(t) = (1-2t)^-5
    What is P[W > 11.78]?

    I calculate the E[X] (mean) = M_w'(0) = 10
    However, I need to know more than this about the distribution than
    simply calculating moments, don't I?

    I looked at this table in Wikipedia ... http://en.wikipedia.org/wiki/Moment-generating_function
    and the given MGF looks (to me) like a chi-squared(10) (I'm not hot on
    chi-squares yet)

    1) Do U need to convert to a pdf for this problem?
    2) Are there standard tables?
    3) If 1) or 2) not relevant to my problem, nudge me in the right

    Thanks, James.
    Rueful Rabbit, Nov 20, 2011
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  2. Rueful Rabbit

    Herman Rubin Guest

    It is a chi-squared with 10 (2x5) degrees of freedom.
    It is not difficult to integrate the density.

    In this case, a few moments work well, as it is approximately normal.
    The formula you quote is for the density; there is another for
    the cdf. However, that is for numerical work.

    In general, inverting a moment generating function, unless
    it is that of a known distribution, is only possible numerically,
    and is not often done.
    Herman Rubin, Nov 21, 2011
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