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FinalExam_F13 - Preliminary Exam Probability 9:00am 2:00pm...

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Preliminary Exam: Probability 9:00am - 2:00pm, August 23, 2013 The exam lasts from 9:00am until 2:00pm. Your goal on this exam should be to demonstrate mastery of probability theory and maturity of thought. Your arguments should be clear, careful and complete. The exam consists of six main problems, each with several steps designed to help you in the overall solution. If you cannot justify a certain step, you still may use it in a later step. On you work, label the steps this way: (i), (ii),... On each page you turn in, write your assigned code number instead of your name. Separate and staple each main part and return each in its designated folder.
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Problem 1 (14 pts) . Let be a random variable and denote ( ) . Let be independent and identically distributed random variables and denote , a. (8 points) (i). Let be a real number. Prove that for each we have . (ii). Show how to modify the LHS of the inequality so that it will hold, with the same RHS, for .
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