First, since the Xi are independent, we could say E(Xi^2) = E(Xi)*E(Xi) = 0*0 = 0. Second, we could also say V(Xi) = E(Xi^2) - [E(Xi)]^2 = E(Xi^2) - 0 = E(Xi^2) in which case E(Xi^2) = V(Xi) = sigma^2.

From these two, we are led to conclude that sigma must be 0 (in which case this is a uniform distribution, not normal). Have I made some error in thinking or is this problem flawed?