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Math Help - moment generation function

  1. #1
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    moment generation function

    Need help solving this
    The moment generating function of a random variable is given by
    M_{X}(t)=(1/6)e^{t}+(1/3)e^{2t}+(1/2)e^{3t.}
    a) Find the distribution function of x.
    b) Find E(X).
    any input highly appreciated.
    thx
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  2. #2
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    The moment generating function of X is the function of t given by the expectation of \exp(tX). Expanded as a power series in t, the coefficient of t^r is m_r / r! where m_r = {\mathbf E} X^r is the r-th moment.

    So the expectation of X, which is m_1, is the coefficient of t in M_X(t) or alternatively M'(0).

    To recover X we see that the random variable which takes values 1, 2, 3 with probabilities 1/6, 1/3, 1/2 respectively has the required MGF.

    There's a good article in MathWorld.
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  3. #3
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    thanks rgep
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