Probability theory: minimizing expectation Let X and Y be two random variables with E(Y)=u and EY^2

Probability theory: minimizing expectation

Let X and Y be two random variables with E(Y)=u and EY^2

a) show that the constant c that minimizes E(Y-c)^2 is c=u

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]