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hw3 - Computer Science 340 Reasoning about Computation...

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Computer Science 340 Reasoning about Computation Homework 3 Due at the beginning of class on Wednesday, October 10, 2007 Problem 1 Suppose you toss three fair, mutually independent coins. Define the following events: A - the event that the first coin is heads B - the event that the second is heads C - the event that the third coin is heads D - the event that the number of heads is even Determine the maximum value of k such that these events are k -wise independent. Problem 2 Let X be a random variable taking integral nonnegative values, Let E [ X ] be its expec- tation, and let E [ X 2 ] denote the expectation of its square. Prove that Pr( X > 0) E [ X ] 2 E [ X 2 ] Hint: You may find the Cauchy–Schwarz inequality useful, which asserts: i a i b i i a 2 i 1 / 2 i b 2 i 1 / 2 . If you use the Cauchy–Schwarz inequality, specify what values of a i and b i are. Problem 3 We have two coins: one is a fair coin and the other is a coin that produces heads with probability 3/4. One of the two coins is picked, and this coin is tossed
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