Let X be a random variable with moment-generating function Mx(t), -h < t < h.
(a) Prove that P(X > α) < e~atMx(t), 0 < t < h. (A proof similar to that used for Chebychev's Inequality will work.)
(b) Similarly, prove that P(X < a) < e~atMx(t), -h < t < 0.
(c) A special case of part (a) is that P(X > 0) < EetX for all t > 0 for which the mgf is defined. What are general conditions on a function h(t, x) such that P(X > 0) < Eh(t, X) for all f > 0 for which Eh(t, X) exists? (In part (a), h(t, x) = etx.)