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Math Help - Moment-Generating Function Questions

  1. #1
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    Moment-Generating Function Questions

    1) Let Y have pdf:
    fy(y) = y, between 0 and 1 (inclusive), 2-y, between 1 and 2 (inclusive), and 0 everywhere else.
    Find My(t)

    2) Fined E(Y^4) if Y is an exponential random variable with fy(y) = xe^(-xy), y > 0

    3) Find the variance of Y if My(t) = e^-2t/(1-t^2)

    4) Calculate P(X less than or equal to 2) if Mx(t) = (1/4 + 3/4e^t)^5
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  2. #2
    MHF Contributor matheagle's Avatar
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    1) Let Y have pdf:
    fy(y) = y, between 0 and 1 (inclusive), 2-y, between 1 and 2 (inclusive), and 0 everywhere else.
    Find My(t)

    M_Y(t)=E(e^{Yt})= \int_0^1ye^{yt}dy +\int_1^2(2-y)e^{yt}dy

    NOW, you do the PARTs
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  3. #3
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    For 2), do I find E(Y^4) by doing integral of Y^4 * fy(y)?? I forget what it means when you have to find expectation of something to a certain power.

    And for 3), I'm assuming I have to find expectation first, but how do I use the moment-generating function to do that?
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  4. #4
    MHF Contributor matheagle's Avatar
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    E(Y^4)=\int_{-\infty}^{\infty} y^4f_Y(y)dy

    Find the variance of Y if My(t) = e^-2t/(1-t^2).....

    Differentiate once and plug in 0 for t, that gives you E(Y)
    (THE first moment)
    Differentiate a second time and plug in 0 for t, that gives you E(Y^2)
    (The second moment)
    That's because this function generates all the moments.
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