$ \hat{y}_{\rm MMSE}= E[Y|X=x] $
I'm gonna try and finish this off....
$ \hat{y}_{\rm MMSE}= E[Y|X=x] = 2\int\limits_{0}^{.5}y\, dy + \int\limits_{.5}^{1}y\, dy = \frac{5}{8} $
I'm not sure if this is wrong... sorry if it is
Nicholas B
