Let X and Y be two unknown variables with E(Y)=[tex]\mu[/tex] and EY2 < [tex]\infty[/tex].
a. Show that the constant c that minimizes E(Y-c)2 is c=[tex]\mu[/tex].
b. Deduce that the random variable f(X) that minimizes E[(Y-f(X))2|X] is f(X)= E[Y|X].
c. Deduce that the random variable f(X) that minimizes E[(Y-f(X))2] is also f(X)= E[Y|X].
The Attempt at a Solution
a. E(Y-c)2 = E[Y2-2Y*c+c2]
other ideas: Can I do the derivative w.r.t. Y, set it equal to zero. But there is an expectation operator, how do you take expectation through the operator.
I can't use a posterior mean, the problem did not specify distr.of the r.v.
b, c. no ideas yet