316_F07_HW1 - today with probability 0.4; and in any other...

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IOE 316 Fall 2007 – Homework 1 Due October 30, 2007 1. (20 points) Three white and three black balls are distributed in two urns in such a way that each contains three balls. We say that the system is in state i, i = 0 , 1 , 2 , 3, if the first urn contains i white balls. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. Let X n denote the state of the system after the n th step. Explain why { X n , n = 0 , 1 , 2 , ... } is a Markov chain and calculate its transition probability matrix. 2. (20 points) Suppose that whether or not it rains today depends on previous weather conditions through the last three days. (a) (10 points) Show how this system may be analyzed by using Markov chain. How many states are needed? List your states. (b) (10 points) Suppose that if it has rained for the past three days, then it will rain today with probability 0.7; if it did not rain for any of the past three days, then it will rain
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Unformatted text preview: today with probability 0.4; and in any other case the weather today will, with probability 0.8, be the same as the weather yesterday. Determine P for this Markov chain. 3. (30 points) Let X and Y be two discrete random variables with joint density function given by y 1 2 3 1 1 12 1 6 1 12 x 2 1 6 1 4 1 12 3 1 12 1 12 Compute the probability of the following events (a) (5 points) X is less than 2.5. (b) (5 points) X is even. (c) (10 points) XY is even. (d) (10 points) Y is odd, given that X is odd. 1 4. (30 points) The joint probability mass function of X and Y , p ( x, y ), is given by p (1 , 1) = 1 9 p (2 , 1) = 1 3 p (3 , 1) = 1 9 p (1 , 2) = 1 9 p (2 , 2) = 0 p (3 , 2) = 1 18 p (1 , 3) = 0 p (2 , 3) = 1 6 p (3 , 3) = 1 9 (a) (20 points) Compute E [ X | Y = i ] for i = 1 , 2 , 3. (b) (10 points) Are the random variables X and Y independent? Explain your answer. 2...
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316_F07_HW1 - today with probability 0.4; and in any other...

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