hw4 - An urn contains n balls numbered 1 ,... ,n . We...

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Homework # 4, Due Tuesday, February 7 1. Problem. Let F and G be distribution functions of random variables. Is it true that the functions 1 2 F + 1 2 G and FG are also distribution functions of some random variables? Please explain. 2. Problem. I try to open the door of my oFce with one of the n similar looking keys in my pocket; one of them is the correct key, the other n - 1 will not turn. Let X be the number of attempts necessary if I choose keys at random from my pocket and drop those that fail to the ground. Let Y be the number of attempts necessary if I choose keys at random from my pocket and put those that fail back in my pocket. ±ind the expectations of X and Y . 3. Problem.
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Unformatted text preview: An urn contains n balls numbered 1 ,... ,n . We remove k balls at random (without replacement) and add up their numbers. ind the expected value of the sum. 4. Problem. We toss a fair coin a number of times and record the result as a sequence of H and T . We say that a run of 3 heads occurs if we get a pattern ...HHH . .. at some point. or example, the sequence THHHHTHHH contains 3 runs of 3 heads. Suppose we toss a coin n times. What is the expected number of runs of 3 heads? 5. Problem. Let X be the number of times we toss a fair coin till the rst run of 3 heads occurs. ind the expectation of X . 1...
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