Assume that X has moment generating function M(t), and define fi(t) = logM(t) show that fi\'\'(t)|t = 0 = Var(x) Solution fi(t) = logM(t) fi \'(t) = M\'(t)/M(t) fi \'\'(t) = (M\'\'(t)M(t)-(M\'(t))^2)/(M(t))^2 fi \'\'(0) =(M\'\'(0)M(0)-(M\'(0))^2)/(M(0))^2 = E(X)-(E(X)^2) =var(X).