Decomposing the average squared distance between X and E(X|Y)

probability statistics random-variables

Solution 1:

Let $E[X|Y]:=g(Y),$ and $\rho=\text{corr}(X,Y).$ Then $$E[ (X-g(Y))^2]=\underbrace{E[X^2]}_{=1}+E[g(Y)^2-2Xg(Y)]\\ =1-\rho^2-E[-\rho^2-g(Y)^2+2Xg(Y)]. $$

Not sure of the significance though...

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