471final - FINAL EXAM, Math 471 (1) [11 pts] Give an...

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FINAL EXAM, Math 471 (1) [11 pts] Give an example of a sequence of random variables { X n } that converges in probability, but not with probability one. Be explicit about what the event space S is and how X i maps S to the real numbers. Also, be sure to prove that your example indeed converges in probability and doesn’t converge with probability one. (2) [9 pts] One has 100 lightbulbs whose lifetimes are independent exponential random variables with mean 5 hours. If the bulbs are used one at a time, with a failed bulb replaced immediately with a new one, what is the approximate probability that there is still a working bulb after 525 hours? (3) [10 pts] Suppose { A n } is a sequence of events in the event space S . Let B be the set of all outcomes ω that belong to infinitely many of the A i . The Borel-Cantelli Lemma says that P ( B ) = 0 if i P ( A i ) < . Use the Borel-Cantelli Lemma to prove that a sequence of random variables { X n } converges to Y with probability one if
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