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Math Help - conditional probability

  1. #1
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    conditional probability

    construct three random variables X Y Z so that

    E(X|Y=y) , E(X|Z=z) are both constant whilst E(X|Z=x,Y=y) isnt a constant??

    not really sure where to go with this??
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  2. #2
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    construct three random variables X Y Z so that

    E(X|Y=y) , E(X|Z=z) are both constant whilst E(X|Z=x,Y=y) isnt a constant
    I assume you typoed and you meant to say
    (X|Y=y) , E(X|Z=z) are both constant whilst E(X|Z=z,Y=y) isnt a constant


    Let Y,Z be independant bernoulli variables. with p=0.5 for both.

    Define:
    X=-1^{Y+Z}

    E(X|y=0) = E(-1^Z) = 0.5*1 + 0.5*-1 = 0
    E(X|y=1) = E(-1^{1+Z}) = 0.5*-1 + 0.5*1 = 0
    So E(X|Y) = 0


    E(X|z=0) = E(X|z=1) =0~~ (by analogy)

    But consider E(X|Y,Z)
    E(X|y=1,z=0) = -1
    E(X|y=1,z=1) = 1
    ...etc. Obviously not constant so i wont do the other two cases.

    That establishes the required properties. Not bad for an economist, eh?
    Last edited by SpringFan25; June 20th 2010 at 02:55 PM.
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