(i) For any r v X, show that P( X ( c) ( X c (c 0), by using the obvious inequality X ( c) In particular, if X ( 0, then P(X ( c) ( X c (ii) For any r v X and any c 0, show that P(X ( c) ( e tc etX (t 0), P(X ( c) ( e tc etX (t 0) (iii) If X2 (, the show that P( X 0) ( ( X )2 X2 In particular, if X ( 0, then P (X 0) ( (X)2 X2

Question: (i) For any r.v. X, show that P(|X| ( c) ( |X|/c (c > 0), by using the obvious inequality |X| ( c). In

(i) For any r.v. X, show that P(|X| ( c) ( ε |X|/c (c > 0), by using the obvious inequality |X| ( c). In particular, if X ( 0, then P(X ( c) ( εX/c?
(ii) For any r.v. X and any c > 0, show that
P(X ( c) ( e-tc εetX (t > 0), P(X ( c) ( e-tc εetX (t < 0).
(iii) If εX2 < (, the show that P(|X| > 0) ( (ε|X|)2 / εX2. In particular, if X ( 0, then P (X > 0) ( (εX)2 / εX2.

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