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# rec07-1 - p X,Y x,y(d Find the marginal PMF p Y y(e Find...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2008) Recitation 7 1 September 25, 2008 1. Random variables X and Y can take any value in the set { 1 , 2 , 3 } . We are given the following information about their joint PMF, where the entries indicated by a * are left unspecified: 3 2 1 y 1/12 1/12 * 2/12 * 1 2 3 x 1/12 2/12 0 * (a) What is p X (1)? (b) Compute the conditional PMF of Y given that X = 1. (c) What is E [ Y | X = 1]? 2. Joe Lucky and the lottery . Joe Lucky plays the lottery on any given week with probability p , independently of whether he played on any other week. Each time he plays, he has a probability q of winning, again independently of everything else. During a fixed time period of n weeks, let X be the number of weeks that he played the lottery and Y the number of weeks that he won. (a) What is the probability that he played the lottery any particular week, given that he did not win anything that week? (b) Find the conditional PMF p Y | X ( y | x ). (c) Find the joint PMF

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Unformatted text preview: p X,Y ( x,y ). (d) Find the marginal PMF p Y ( y ). (e) Find the conditional PMF p X | Y ( x | y ). Do this algebraically using previous answers. (f) Based on an intuitive interpretation of part (a), rederive your answer to part (e). 1 Published September 23, 2008 Page 1 of 2 Massachusetts Institute of Technology Department of Electrical Engineering &amp;amp; Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2008) 3. Problem 2.32, page 128 of the text. D. Bernoullis problem of joint lives. Consider 2 m persons forming m couples who live together at a given time. Suppose that at some later time, the probability of each person being alive is p , independently of other persons. At that later time, let A be the number of persons that are alive and let S be the number of couples in which both partners are alive. For any number of total surviving persons a , nd E [ S | A = a ]. Page 2 of 2...
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