Problems in Probability

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Fall 2011 Strategic Practice 10: Section 2 (Inequalities) - Question 2
(a) Show that E(1/X) > 1/(EX) for any positive non-constant r.v. X.
(b) Show that for any two positive r.v.s X and Y with neither a constant multiple of the other, E(X/Y)E(Y/X) > 1.
Solution: (a) The function g(x) = 1/x is strictly convex because g''(x) = 2x^−3 > 0 for all x > 0, so Jensen’s inequality yields E(1/X) > 1/(EX) for any positive non-constant r.v. X. (b) The r.v. W = Y/X is positive and non-constant, so (a) yields E(X/Y) = E(1/W) > 1/E(W) = 1/E(Y/X).
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