# Math Help - MMSE estimator

1. ## MMSE estimator

I need some help here...

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

Then I need to find
1) Best predictor of $X$ given $Y$, and of $Y$ given $X$.

So for this I do:
$\hat{X_{MMSE}}(y) = E(X|Y) = E(Y^2|Y) = y^2 E(1|Y) = y^2$
$\hat{Y_{MMSE}}(x) = ??$ (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.

2. For point 2), I found how to do it.
It's described here, pages 87-88.

However, still don't know how to find $\hat{Y_{MMSE}}(x) = E(Y|X)$.