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hw3 - win by drawing his/her own number – player 1 wins...

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Homework 3 due Monday, 7/25/2011 Note: You should show all work on each question to get full credit. 1 Suppose X and Y are independent, integer-valued random variables such that P [ X = n ] = P [ Y = n ] = q n - 1 p for n 1 (assume p, q (0 , 1), p + q = 1). (a) [10pts] Let S = X + Y . Find the distribution on S (that is, a formula for P [ S = n ]). (b) [10ts] Let M = min ( X, Y ). Find the distribution on M . 2 [14pts] We have a box containing 21 marbles, each bearing one of the integers 1 – 6, so that the number 1 appears once in the box, the number 2 appears twice, . . . , the number 6 appears 6 times. Consider the following game played in turns by 6 players: The players take turns in order until someone wins, starting with player 1, then player 2, etc. On his / her turn, a player draws one marble, with replacement. Each player can
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Unformatted text preview: win by drawing his/her own number – player 1 wins by drawing a 1, player 2 wins by drawing a 2, etc. • The game stops when someone wins, otherwise it keeps going. The marbles are drawn with replacement , so the game can go on for arbitrarily many turns. What is the probability that player 3 wins this game? 3 Suppose the number of telemarketing calls reaching my house during any given day (24-hours) is Poisson with parameter 5. (a) [10pts] What is the probability that at least 4 calls reach my house during the month of June? (b) [12pts] What is the probability that there are exactly 4 days during the month of June during which I get at least one call? Last compiled at 7:50 P.M. on July 19, 2011...
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