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A Parameterizing conditional expectations (Gaussian case)

  1. Mar 2, 2016 #1
    Consider three jointly normally distributed random variables X,Y and Z.

    I know that in the Gaussian case E[Z | X,=x Y=y]=xßZX;Y +yßZY;X
    where ßZX;Y notes the regression coefficient of Z on X conditional on Y (and ßZY;Xis analogously defined).

    Is the following derivation correct?

    E[Z| X>x, Y<y] = E [ E[Z | X,Y] | X>x, Y<y]
    ZX;Y E[X | X>x, Y<y]+ßZY;X E[Y |X>x, Y<y]
  2. jcsd
  3. Mar 2, 2016 #2


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    Yes it is correct.
    The first equality is an application of the Tower Law.
    The second equality actually comprises two steps:
    1. substitute your formula for E[Z | X,Y] from above for the inner expectation
    2. Use the linearity of the expectation operator and the fact that the betas are constants to get the result.
  4. Mar 10, 2016 #3
    Great! Thanks a lot for spelling it out
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