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Unformatted text preview: Math 3225 Final Exam, Fall 2005 December 13, 2010 1. Define the following a. A sigma algebra. b. The Kolgomorov axioms of probability. c. The Law of Large Numbers. d. The Central Limit Theorem. 2. Bob wishes to transmit one bit of information across a noisy channel to Alice. If the bit to be transmitted is 1, then Bob sends three 1s in a row; and, if the bit to be transmitted is 0, then Bob sends three 0s. Alices rule for determining which bit Bob wanted her to receive is Majority Rules: If the number of 1s among the three received bits is 2 or 3, then Alice reports that Bob was trying to send her a 1, and if the number of 0s among the three received bits is 2 or 3, then Alice reports that Bob was trying to send her a 0. Suppose that the channel makes an error 1/3 of the time (thus, if you sent 1000 1s in a row, about 333 of them will be flipped to 0s, and whether a bit is flipped is independent of whether the other bits get flipped). And suppose that Bob chooses the bit 0 or 1 of information with equal probability. If Alice recieves the noisy message 110, what is the probability that Bobs intended bit was, indeed, 1, as Alice reports?bit was, indeed, 1, as Alice reports?...
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