Simple question here (I think). I have to find $E(X\mid Y)(y)$ where $X$ is the value of the first roll and $Y$ is the sum of the two dice. Normally, I wouldn't have any trouble with this sort of problem, but the definitions of the two variables are "flipped," so to speak, making it a little confusing for me. I know that
$E(X|Y)(y) = \sum_x{xP(X\mid Y)}=\frac{\sum_xxP(X=x, Y=y)}{P(Y=y)},$
but this doesn't really get me anywhere. From that summation, I get
$E(X|Y)(y) = \frac{1\cdot P(1, Y=y)}{P(Y=y)} + \frac{2\cdot P(2, Y=y)}{P(Y=y)} + \dots + \frac{6\cdot P(6, Y=y)}{P(Y=y)}.$
Here's where I'm stumped. How can I put $E(X|Y)(y)$ in terms of just $y$?
Thanks!
EDIT: To be clear, I need to find the expected value of the first roll given the sum of the two dice.