Computer Science 174 - Spring 2000 - Canny - Final Exam

# Computer Science 174 - Spring 2000 - Canny - Final Exam -...

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CS174 Final Exam Solutions Spring 2000 J. Canny May 17 1. (20 points) (a) (4 points) What is the probability that a random permutation of has at least one ﬁxed point? Assume is large. The number of ﬁxed points in a random permutation has Poisson(1) distribution. Hence the probability of no ﬁxed points is and the probability of at least one ﬁxed point is . (b) (4 points) If where , express in terms of , and . Distributing the expression, we get: And the last expression simpliﬁes to . (c) (4 points) Suppose you draw a random card from a pack of cards, look at it, replace it in the pack, and then reshufﬂe. Repeat times. What is the expected value of until you have seen all the cards? This is the coupon collector’s problem, so we should expect to draw cards. (d) (4 points) Suppose you want to bound the probability of more than heads in tosses of a fair coin. Out of Markov, Chebyshev and Chernoff, which bound would you use and why? Assuming is large, you shouldnt need to compute anything to answer this question. Since all the tosses are independent, we can use Chernoff bounds which are the strongest bounds of the three. (e) (4 points) Suppose you draw cards at random from a deck of cards, with replace- ment. How large should be in terms of so that you have small probability (say less than half) of drawing the same card twice?

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## Computer Science 174 - Spring 2000 - Canny - Final Exam -...

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