416-3 - (d If you win $1 each time head shows up and if you...

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Denker SPRING 2010 416 Stochastic Modeling - Assignment 3 Due date: Monday, 02/08/2010 in class Problem 1: (Problem 46, p.173) Show that Cov ( X, Y ) = Cov ( X, E [ Y | X ]) . Problem 2: (Problem 49, p. 174) A and B play a series of games with A winning each game with probability p . The overall winner is the Frst player to have won two more games than the other. (a) ±ind the probability that A is the overall winner. (b) ±ind the expected number of games played. Problem 3: (Problem 50, p. 174) There are three coins in a barrel. These coins ²ipped, will come up heads with respective probabilities 0 . 3, 0 . 5, and 0 . 7 respectively. A coin is randomly selected from among these three and is then ²ipped ten times. Let N be the number of heads on the ten ²ips. ±ind (a) P ( N = 0). (b) P ( N = n ) for n = 1 , 2 , 3 , ..., 10. (c) Does N have a binomial distribution?
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Unformatted text preview: (d) If you win $1 each time head shows up and if you lose 1$ if tail shows up, are you playing a ’fair’ game? Explain your reasoning. Problem 4: (Problem 56, p.175) The number of daily accidents in a certain city is Poisson distributed. On a rainy day the parameter is λ = 0 . 8 and on a dry day it is μ = 0 . 2. Let X denote the number of accidents tomorrow, when it is predicted that it is a rainy day with probability 0 . 6. ±ind (a) E [ X ]. (b) P ( X = 0). (c) V ar ( X ). Problem 5: (Problem 57, p.175) The number of storms in an upcoming season is Poisson distributed with parameter λ which itself is random and has a uniform distribution over the interval (0 , 5). ±ind the probability that there are at least 3 storms this season....
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