final#1 - ORIE 360/560 { Engineering Probability and...

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Unformatted text preview: ORIE 360/560 { Engineering Probability and Statistics II Fall 2003 FinalExam All problems have equal weight. SHOW YOUR WORK. GOOD LUCK! ables, X with parameter 1, and Y with parameter 2. Let V = X + Y and W = X=Y . (a) Find the joint density of V and W . (b) Are V and W independent (try answering this question even if you are not sure your answer to part (a) is correct). density function Problem 1 Let X and Y be independent exponential random vari- Problem 2 Random variables X and fX;Y (x; y ) Y have a joint probability 1 = c(y (a) Find c. (b) Find the marginal densities of X and Y . (c) Find the conditional density of Y given X = x and the conditional density of X given Y = y. (d) Compute the conditional mean and the conditional variance of X given Y = y for 0 < y < 1. 0 x) if 0 x y otherwise. Problem 3 A sample X ; X ; : : : ; Xn comes from a continuous distribution with the density fX (x) = ( + 1)x ; 0 < x < 1 for some unknown parameter > 1. (a) Compute the maximum likelihood estimator of . (b) Compute the moment estimator of . (c) Compute the Bayes estimator of that maximizes the posterior density if the prior density of is p( ) = e for > 1. ( +1) 1 2 and X , observed on two successive days follow a bivariate normal distribution with parameters = = 75, = = 8 and = :9. (a) Find the probability that the average over two successive days of the daily maximal air temperatures exceeds 80. 2 1 2 1 2 Problem 4 Suppose that the daily maximal air temperatures X 1 1 (b) Find the probability that the maximal air temperature on the second day exceeds 80 given that the maximal air temperature on the rst day was equal to 80. Problem 5 Let X ; : : : ; Xn be iid Pareto random variables with the density fX (x) = 5x for x > 1. (a) Find the pdf of the largest observation Y = max(X ; : : : ; Xn). (b) Find the pdf of the smallest observation Z = min(X ; : : : ; Xn). 1 6 1 1 2 ...
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final#1 - ORIE 360/560 { Engineering Probability and...

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