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Math Help - Conditional Expectation

  1. #1
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    Conditional Expectation

    Hi,
    The problem is in the attached PDF file.

    Thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ganiba View Post
    Hi,
    The problem is in the attached PDF file.

    Thanks.
    I hate having to follow a link to see the problem.

    RonL
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  3. #3
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    Quote Originally Posted by ganiba View Post
    Hi,
    The problem is in the attached PDF file.

    Thanks.
    I'm not sure where this will go, but lets start by expanding the integrals, as X and Y are independent:

    E_Y\left[ E_X \left[ \chi_{\left\{ X>\alpha Y\right\}} X \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha y}^{\infty} x \ p_X(x)\ p_Y(y)\ dx dy

    and:

    E_X\left[ E_Y \left[ \chi_{\left\{ Y>\alpha^{-1} X\right\}} Y \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha^{-1} x}^{\infty} y \ p_X(x)\ p_Y(y)\ dy dx

    Now we can observe that the region over which we are integrating in the first integral is the part of the x-y plane below y=\alpha^{-1} x and that in the second integral is that part of the plane above the same line.

    Quite where to go from here I don't know

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    I'm not sure where this will go, but lets start by expanding the integrals, as X and Y are independent:

    E_Y\left[ E_X \left[ \chi_{\left\{ X>\alpha Y\right\}} X \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha y}^{\infty} x \ p_X(x)\ p_Y(y)\ dx dy

    and:

    E_X\left[ E_Y \left[ \chi_{\left\{ Y>\alpha^{-1} X\right\}} Y \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha^{-1} x}^{\infty} y \ p_X(x)\ p_Y(y)\ dy dx

    Now we can observe that the region over which we are integrating in the first integral is the part of the x-y plane below y=\alpha^{-1} x and that in the second integral is that part of the plane above the same line.

    Quite where to go from here I don't know

    RonL
    Thanks Ront.
    I attached the file because i don't know how to include mathematical formulaes in the Forum, so thanks again for the help.

    I want to add that X and Y are POSITIVE random distributions, so integral starts from 0 to infinity.

    Thanks.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by ganiba View Post
    Thanks Ront.
    I attached the file because i don't know how to include mathematical formulaes in the Forum, so thanks again for the help.

    I want to add that X and Y are POSITIVE random distributions, so integral starts from 0 to infinity.

    Thanks.
    But that does not realy help, its a idea that's needed now.

    ronL
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