Consider the problem of approximating a continuous analog signal $X$ with a discrete digital signal $Y$. To do this, we may divide the range of $X$ into discrete intervals, $[a_0,a_1)$, $[a_1,a_2)$, etc. Then, if $X$ takes on a value in the interval $[a_0,a_1)$, we assign the value $y_1$ to the discrete signal (predictor) $Y$, and so on for the other intervals.

Suppose that $X$ is distributed on $[0,1]$ according to the PDF $f(x) = 1.6 x + 0.2$. If we discretize by picking $a_0 = 0$, $a_1 = 1/2$ and $a_2 = 1$, how should we pick $y_1$ and $y_2$ to generate the best predictor of $X$ (that is, to minimize $E[(X-Y)^2]$?

$y_1 =$
$y_2 =$

In this case, what is the expectation?
$E[(X-Y)^2] =$