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Math Help - X, Y iid, need moment generator function of sum and product

  1. #1
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    Exclamation X, Y iid, need moment generator function of sum and product

    Can someone give me some direction on this? I tried something but I have no definitions or theorems to support whether what I did is correct. Thanks!

    Let X~Normal( \mu, \sigma), Y~Gamma( \alpha, \beta), and X and Y are independent. The moment generating function of X and Y are

    m_{X}(t) = e^{\mu t+\sigma^2 t^2 /2} and m_{Y}(t) = (1- \beta t)^\alpha, respectively.

    Find the moment generating function of X+Y.
    Find the moment generating function of 5X.
    Find the moment generating function of 2 + Y.
    Find the moment generating function of 2 + 3X + 4Y.

    Thanks!
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  2. #2
    MHF Contributor matheagle's Avatar
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    I'll do the last one.

    The MGF of 2 + 3X + 4Y, by definition is

     E(e^{(2 + 3X + 4Y)t})=e^{2t} E(e^{3Xt})E(e^{4Yt})

    by independence and algebra. Now combine the constants with the t's....

    =e^{2t} E(e^{X(3t)})E(e^{Y(4t)})

    =e^{2t} M_X(3t)M_Y(4t)

    Finally plug in the 3t where you see a t into the MGF of X
    and plug in the 4t where you see t in the MGF of Y.
    Last edited by matheagle; November 17th 2009 at 04:12 PM.
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