In Exercise 1, we assumed that there could be at most one distribution with moments E(Xk) = 1/3 for k = 1, 2, . . . . In this exercise, we shall prove that there can be only one such distribution. Prove the following facts and show that they imply that at most one distribution has the given moments.
a. Pr(|X| ≤ 1) = 1. (If not, show that limk→∞ E(X2k) = ∞.)
b. Pr(X2 ∈ {0, 1}) = 1. (If not, prove that E(X4) < E(X2).)
c. Pr(X =−1) = 0. (If not, prove that E(X) < E(X2).)