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Math Help - Moment generating function problem

  1. #1
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    Moment generating function problem

    Hello this is my first post in this forum. I hope I can give back some help in my spare time. I'll go on with my question.

    Given the moment generating function M_x (t) = e^{3t + 8t^2 } , find the moment generating function of the random variable Z = \frac{1}<br />
{4}(X - 3), and use it to determine the mean and the variance of Z.

    I am a quite confused since I don't know exactly what are they asking from me. The only thing that closely resembles an answer that I have come up with is to calculate \mu and \sigma ^2 using the given generating function and use the standard deviation obtained as the integration limits for finding the mgf of Z. If my approach ok? I'm i totally wrong?

    Edit: I should probably add that I calculated the mean and sd and arrived at a mean of 3 and sd of 4. This just doesn't seem right to me.

    edit 2: Please disregard. The answer is here
    Last edited by mr fantastic; April 22nd 2009 at 05:35 AM. Reason: Clarified edit 2 made by the OP
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  2. #2
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    moment generating function

    Set ϕ(t)=M_{X}(t)=Ee^{tX} and ψ(t)=M_{Z}(t)=Ee^{tZ}=Ee^{t((X/4)-(3/4))}=Ee^{(t/4)X}e^{-((3t)/4)}=e^{-((3t)/4)}Ee^{(t/4)X}=e^{-((3t)/4)}ϕ((t/4))=e^{-((3t)/4)}e^{3(t/4)+8((t)/(16))}=e^{((t)/2)}.
    We have ψ′(t)=te^{((t)/2)} and ψ′′(t)=e^{((t)/2)}+te^{((t)/2)}, and so EZ=ψ′(0)=0, EZ=ψ′′(0)=1 and VarZ=EZ-(EZ)=1


    Best regards, Aurel Spataru
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