Let X and Y be two random variables with E(Y)=u and EY^2 < infinity.
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]
This solution helps explore how to minimize expectation within the context of probability theory.