Deduce that the random variable f(X) that minimizes E[(Y-f(X))^2] is f(X)= E[Y|X].

Answer 1

To rephrase, you want to prove that the the Function f(X) that minimises the expected value of (Y - f(x))² is the expected value of Y given X. (i.e. E[Y|x] ) This is to me intuitive as the smallest average delta for any given X of the function Y-f(x) would be where f(x) = expected value of Y (given x).

Since E[Y|X] is a constant (and equal to f(X)) and given that E[ Y + c] = E[Y] + c (where c is a constant)

then E[Y-f(X)]² = E[Y|x]² - E[Y|x]² or 0

This is just a shortcut proof.