More precisely, suppose we have the following statements for a random variable X: (A) There is a > 0 such that E[ex] < +∞; Az (B) There is X>0 and m >0 such that P[|X| ≥ x] ≤ me¯ for all x ≥ 0. We show that (A) implies (B) (with m := E[eª|X] and À ≤ a in (B)) and that (B) implies (A) (with a in (A) being any number a € (0, A)), so that statement (A) and (B) are equivalent. (a) (A implies B) Suppose X is a random variable for which m := E[eª|X ] < +∞, where a > 0. Suppose ≥ 0 and define A(x) = {w: X(w) ≥ x} Show for any >> 0 that A(x) = {w: e^|x (w)| ≥ e^x}. (b) By considering the cases when w€ A(x) and w€ Ã(x), show that ex> e IA(z).

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