We know that if $E(X|Y) = E(X)$, then $X$ and $Y$ are uncorrelated. I found out that the reverse statement is not always true, ie. $X$ and $Y$ are uncorrelated, but $E(X|Y)$ is not equal to $E(X)$. But I don't get why... If I am right, can you give an example where this happens??
2026-04-01 21:57:53.1775080673
Random variables, Correlation and expectation
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$Y\sim N(0,1)$ and $X=Y^{2}$. Note that $E(X|Y)=Y^{2}$ and $EX=1$. Now $E(XY)=EY^{3}=0=(EX) (EY)$.