a5__ - X be the sum of numbers on the two dice 1 What is E...

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Cmpt-405 July 27, 2011 Assignment 5 Problem 1. View each route in Phase 1 of the permutation routing algorithm for Hyper- cubes as directed path from the source to an intermediate destination. Prove that once two routes separate they do not rejoin. Problem 2. Does the statement in Problem 1 imply that for any two packets v i and v j , there is at most one queue q such that v i and v j are in the queue at the same step? Problem 3. Suppose m balls are thrown into n bins. Give the best bound you can on m to ensure that the probability of there being a bin containing at least two balls is at least 1 / 2. Problem 4. Give examples of functions f and random variables X where 1. E [ f ( X )] f ( E [ X ]), 2. E [ f ( X )] = f ( E [ X ]), and 3. E [ f ( X )] f ( E [ X ]). Problem 5. Suppose that we independently roll two standard dice. Let X 1 be the number that shows on the first die, and X 2 the number that shows on the second die. Let
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Unformatted text preview: X be the sum of numbers on the two dice. 1. What is E [ X | X 1 is even]? 2. What is E [ X | X 1 = X 2 ]? 3. What is E [ X 1 | X = 9]? 4. What is E [ X 1-X 2 | X = k ] for k in the range [2 , 12]? Problem 6. • For two probabilistic events A and B , does A ∩ B = ∅ imply independence of A and B ? • Give an example where Chernov bounds only give the trivial answer, that is, the obviously correct but not terribly useful p ( X ≥ (1 + ± ) E [ X ]) ≤ 1. • Suppose that n balls are tossed into n bins, where each toss is independent and the ball is equally likely to end up in any bin. What is the expected number of empty bins? What is the expected number of bins with exactly one ball? 1...
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