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Math Help - stupid normal distribution question

  1. #1
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    Computing conditional expectation...

    if I have X1,X2 iid normal with N(0,1)

    and I want to find E(X1*X2 | X1 + X2 = x)

    Can I simply say: X1 = x - X2 and thus

    E(X1*X2 | X1 + X2 = x) =

    E[ (x - X2)*X2) = E[ (x * X2) - ((X2)^2) ] <=>

    x*E[X] - E[X2^2] =
    0 - 1 =
    -1

    ???
    Last edited by sezmin; February 24th 2010 at 08:54 AM. Reason: better wording
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  2. #2
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    Also,

    I think I heard somewhere that if you have X and Y are two random variables, both of which have the standard normal distribution N(0,1), then X and Y are independent...

    This sounds strange to me, is it true?
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  3. #3
    MHF Contributor matheagle's Avatar
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    no and no
    you need the covariance to be zero in the second question.
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  4. #4
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    i see, how do i find the first one then?

    i see, for the second one, thanks!
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  5. #5
    MHF Contributor matheagle's Avatar
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    I'm bet there is a clever way to do the first, but the straight fowrard way will work.

    Obtain the conditional density.
    let W=XY and Z=X+Y.
    From the joint density of X and Y you can obtain the joint density of W and Z.

    Then f(W|Z)={f(W,Z)\over f(Z)}

    So E(W|Z)=\int wf(w|Z)dw

    The marginal density of Z is obvious.
    Last edited by matheagle; February 24th 2010 at 03:22 PM.
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