Exsheet3 - (b) X n → X and X n → Y almost surely. Show...

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EXAMPLE SHEET 3 CLASSICAL PROBABILITY, 2008, J. MORIARTY 1. Assume that X n , n = 1 , 2 , ... are random variables on some proba- bility space such that X n c in distribution, where c is some constant. Show that X n c also in probability. 2. Prove the following implications: i) X n X a.s. Y n Y a.s. b = X n + Y n X + Y a.s. ii) X n X in r th mean Y n Y in r th mean b = X n + Y n X + Y in r th mean iii) X n X in probability Y n Y in probability b = X n + Y n X + Y in probability. 3. Assume that (a) X n X and X n Y , both in probability. Show that P ( X = Y ) = 1.
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Unformatted text preview: (b) X n → X and X n → Y almost surely. Show that P ( X = Y ) = 1. (c) X n → X and X n → Y in r th mean for some r ≥ 1. Show that P ( X = Y ) = 1. 4. Give examples of two (non-constant) random variables X and Y such that (a) H¨older’s inequality reduces to an equality for some p > 1 and q > 1 (b) Minkowski’s inequality reduces to an equality for some p ≥ 1. 1...
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This note was uploaded on 03/15/2010 for the course STATISTIC 472 taught by Professor Amjad during the Spring '08 term at Yarmouk University.

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