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If i have 2 random vectors $X$ and $Y$, each of them with own different pdf, what is the conceptual interpretation of saying that the conditional expectation of $X$ given $Y$ is null? i.e. \begin{equation} E[X|Y] = 0 \end{equation} Thank you very much for your attention.

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I would prefer to see something like "$E[X|Y=y]=0$ for all $y$", which could be taken as meaning that whatever the value of $Y$, the expected value of $X$ is $0$.

One example of this might be $Y$ having any distribution you want and $X$ taking the values $\pm Y$ with probability $\frac12$ each. Since $|X|=|Y|$, they are not independent even though the value of $Y$ does not affect the expectation of $X$.

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interesting... but i'm still having a doubt: why have you said $|X| = |Y|$ ? what's that identity's meaning? Thank you very much! –  Alfatau Aug 24 '12 at 16:48
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In Henry's example X=Y or X=-Y so |X|=|Y|. –  Michael Chernick Aug 24 '12 at 17:31
    
I probably should have required $Y$ to have a finite mean –  Henry Aug 24 '12 at 22:16
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