(Solution) - i For any r v X show that P X c -(2025 Original AI-Free Solution)

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(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.