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STAT 333 a2 - Stat 333 Assignment 2 Fall 2011 Due Thursday...

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Stat 333 - Assignment 2 - Fall 2011 Due Thursday November 10 at the beginning of class 1. Consider repeated Bernoulli trials, each with a probability of success P ( S ) = p . Let e n = Pr(even number of S ’s in n trials). Assume zero is even. (a) Show that e n = q · e n - 1 + p · (1 - e n - 1 ) , n 1 . (b) Hence, find the probability generating function of { e n } n 0 . (c) Let p = 0 . 5 . Find an explicit expression for e n , n 0 . 2. Prove the Delayed Renewal Relation. (Hint: the proof is very similar to that of the Re- newal Relation, and you have to be very careful with the starting values of sums.) 3. Javid’s world-class cricketing career has recently begun. In a single game of cricket, he is said to score a century if he scores at least 100 runs. The number of matches until he scores a century has a Geometric (0 . 01) distribution. (a) Let λ be the event that he scores a century in a game. i. Explain carefully why λ is a renewal event. ii. Determine the renewal sequence { r n } .
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