MMSE estimator

Player1 · #1 $Old$ October 11th, 2008, 01:53 PM

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.

Player1 · #2 $Old$ October 11th, 2008, 05:08 PM

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

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