midterm solutions - Computer Science 340 Reasoning about...

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Computer Science 340 Reasoning about Computation Homework 6 (take home midterm) Due at the beginning of class on Wednesday, November 7, 2007 There are six problems on the midterm. You must work on them on your own. No collaboration is allowed. Problem 1 Consider a random subset R of { 1 , 2 , . . . , n } created as follows: Initially, R is empty. For each i ∈ { 1 , 2 , . . . , n } , i is included in R with probability p . Here p [0 , 1] is a given parameter, and these choices are mutually independent. (a) Compute the probability that R contains k elements. (b) Compute the probability that R contains an odd number of elements. Note: For full credit, you should give a closed form expression for this probability. A summation is not considered a closed form expression. Hint: If a is the probability that R contains an odd nmber of elements, let b be the prob- ability that R contains an even number of elements. Find a relation between a and b other than the trivial a + b = 1. Solution sketch: The probability that R has size k is n k p k (1 - p ) n - k , since there are ( n k ) such subsets and each has a probability of p k (1 - p ) n - k to be picked. The probability that the size of R is odd is simply k odd n k p k (1 - p ) n - k . Observe that n k =0 n k p k (1 - p ) n - k = ( p + (1 - p )) n = 1 n k =0 ( - 1) k n k p k (1 - p ) n - k = ((1 - p ) - p ) n = (1 - 2 p ) n .
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Subtract the second equation from the first one above. Thus, 2 k odd n k p k (1 - p ) n - k = 1 - (1 - 2 p ) n . Hence the required probability is (1 - (1 - 2 p ) n ) / 2. Problem 2 Suppose we place n points at random locations on a circle of unit circumference. Each point is placed at a location chosen with uniform distribution on the circumference, and the random choices made for the n points are mutually independent. In this problem, you will prove that, with high probability, the gap between any two consecutive points is O ((log n ) /n ). (a) Consider an arbitrary interval I of length c (ln n ) /n on the circumference. Show that the probability that this interval is empty (i.e. containts none of the n points) is at most 1 /n c .
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