I need some help here...

Have 2 r.v.'s $\displaystyle X,\;Y$, such that the PDF of $\displaystyle Y$ is known and $\displaystyle X=Y^2$

Then I need to find

1) Best predictor of $\displaystyle X$ given $\displaystyle Y$, and of $\displaystyle Y$ given $\displaystyle X$.

So for this I do:

$\displaystyle \hat{X_{MMSE}}(y) = E(X|Y) = E(Y^2|Y) = y^2 E(1|Y) = y^2$

$\displaystyle \hat{Y_{MMSE}}(x) = ??$ (the square root confuses me here)

2) Same as 1) but the predictors have to be linear

I don't know how to solve this

Thanks.