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Math Help - Uniform and MGF please help

  1. #1
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    Unhappy Uniform and MGF please help

    1.Find the mean and variance of a uniform distribution [0,1] by MGF.

    2.Y=a+bX1+cX2+d find the MGF of X1 and X2 ?

    Please help
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  2. #2
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    Quote Originally Posted by Jason2009 View Post
    1.Find the mean and variance of a uniform distribution [0,1] by MGF.

    2.Y=a+bX1+cX2+d find the MGF of X1 and X2 ?

    Please help
    2. Makes no sense to me. Perhaps you have not posted the whole question?


    1. First calculate m(t) = E(e^{tX}) = \int_0^1 e^{tx} \cdot 1 dx. Then remember that E(X) = \left. \frac{dm}{dt} \right|_{t = 0} and E(X^2) = \left. \frac{d^2 m}{dt^2} \right|_{t = 0}.
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  3. #3
    MHF Contributor matheagle's Avatar
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    are X1 and X2 uniform (0,1)?

    E(e^{Yt})=E(e^{t(a+bX1+cX2+d)})

    Now using indep (which I assume you have)

     =e^{(a+d)t}E(e^{t(bX1+cX2)}) =e^{(a+d)t}E(e^{btX1+ctX2)})=e^{(a+d)t}E(e^{btX1})  E(e^{ctX2}).

    which is...

    e^{(a+d)t}MGF_{X1}(bt)MGF_{X2}(ct).

    And I don't get the a and d, why two constants?
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  4. #4
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    No X1 and X2 are not uniform distribution .
    The 2nd question is :X1 and X2 are independent variables and Y= a+bX1+cX2+d Find the MGF of Y in terms of MGF of X1 and X2.

    Thanks your help anyway
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  5. #5
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    Quote Originally Posted by Jason2009 View Post
    No X1 and X2 are not uniform distribution .
    The 2nd question is :X1 and X2 are independent variables and Y= a+bX1+cX2+d Find the MGF of Y in terms of MGF of X1 and X2.

    Thanks your help anyway
    You're expected to know and use the following two results:

    1. If U = aX + b then M_U (t) = e^{bt} M_X (at).

    2. If U = X + Y and X and Y are independent random variables then M_U (t) = M_X (t) \cdot M_Y (t).
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