
MMSE estimator
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:
$\displaystyle \hat{X_{MMSE}}(y) = E(XY) = E(Y^2Y) = y^2 E(1Y) = y^2$
$\displaystyle \hat{Y_{MMSE}}(x) = ??$ (the square root confuses me here) (Shake)
2) Same as 1) but the predictors have to be linear
I don't know how to solve this
Thanks.

For point 2), I found how to do it.
It's described here, pages 8788.
However, still don't know how to find $\displaystyle \hat{Y_{MMSE}}(x) = E(YX)$.