Let X be a discrete random variable, all of whose moments are equal to 1 3 . (a)Show that M(t) = {1/3} e^t + 2/3 is one possible moment generating function for X. Then determine the probability mass function for X in this case. (b) Now show that there is in fact only one distribution with this sequence of moments. Do this by proving the following facts, and then conclude that the distribution of X is unique. i. Show that P(|X| ≤ 1) = 1 (because otherwise limn→[infinity] E(X^{2n} ) = [infinity]). ii. Show that P(X^2 ∈ {0, 1}) = 1 (because otherwise E(X^4) < E(X^2)).
