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Consider two random random variables $X$ and $Y$ with finite variance. Is it true that $X= E(Y\vert X)$ iff there is a $+1$ or $-1$ correlation between $X$ and $Y$?

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    $\begingroup$ Well, if the correlation has value $-1$, then it might be that $E[Y\mid X] = -X$ and not $+X$ as you want to prove/ $\endgroup$ – Dilip Sarwate Apr 11 '17 at 2:55
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Simple counterexample: Let $X$ be some random variable (with finite variance) and let $Y=5X$. Then the correlation between $X$ and $Y$ is 1, but we have $$ \DeclareMathOperator{\E}{\mathbb{E}} \E \left[ Y \mid X \right] = \E\left[ 5X \mid X \right] = 5X \not = X $$

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