Consider the problem of approximating a continuous analog si

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, [a0,a1), [a1,a2), etc. Then, if X takes on a value in the interval [a0,a1), we assign the value y1 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)=1.8x+0.1. If we discretize by picking a0=0, a1=1/2 and a2=1, how should we pick y1 and y2 to generate the best predictor of X (that is, to minimize E[(XY)2]?

In this case, what is the expectation?

Solution

the values are y₁ = (1.8*0.5)+0.1 = 1

y₂ = (1.8*1)+0.1 = 1.9

hence E[(XY)²] = 0.3567