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If $E[X|Y=y]=y$, does $E[X]=E[Y]$? Similarly, if $E[X|Y=y]=y^2$, does $E[X]=E[Y^2]$?

I'm having some trouble with this conditional expectation concept, although it seems intuitively true

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    The notation $E[X\mid Y=y]=y$ means that the conditional expectation of $X$ with respect to $Y$ is $Y$. Since $\Omega$ in $\sigma(Y)$, we have $E[X]=\int_{\Omega}E[X\mid Y]dP=\int_{\Omega}YdP=E[Y]$, if these two random variables are integrable.2011-12-18
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    Thank you Davide! Does that also hold for the Y^2 case?2011-12-18
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    $E[X]=\int_{\Omega}E[X\mid Y]dP=\int_{\Omega}Y^2dP=E[Y^2]$, so these results are true if $X$ is integrable and $Y$ square integrable.2011-12-18

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