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# hw6 - Btu and standard deviation 25 million Btu a What can...

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ORIE 3500/5500 – Engineering Probability and Statistics II Fall 2010 Assignment 6 Problem 1 Let X be a standard uniform random variable. Let θ > 0 be a parameter, and Y = X θ - X . Among the following 4 values of θ : θ = 0 . 5 , 1 . 2 , 1 . 5 and θ = 3, find the one that makes the correlation ρ X,Y the largest. Problem 2 Suppose that X and Y are discrete random variables with the joint p.m.f. p X,Y ( i, j ) = 1 / 6 if i, j = 0 , 1 , 2, and i + j 2. ( a ) Try figuring out the sign of the correlation ρ X,Y without calcu- lating it. Explain your reasoning. ( b ) Now calculate the correlation ρ X,Y and check your reasoning in part ( a ). Problem 3 Let X be a continuous random variable with a density f X ( x ) = 1 2 e - x + e - 2 x , for x > 0. ( a ) Compute the moment generating function of X . What is the range on which the moment generating function is defined? ( b ) Use the moment generating function from part ( a ) to compute the mean, the second moment and the variance of X . Problem 4 The number of Btu’s of petrolium and petrolium products used per person in the United States in 1975 had the mean 153 million

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Unformatted text preview: Btu and standard deviation 25 million Btu. ( a ) What can you say about the proportion of the population that used between 103 and 203 million Btu during that year? ( b ) Estimate the proportion of the population that used in excess of 228 million Btu during that year using two diﬀerent estimates learned in class. Which one gives a better bound? Problem 5 Let X be a Pareto random variable, with the density f X ( x ) = 4 x-5 , x ≥ 1 , and equal to zero otherwise. 1 ( a ) Calculate both the exact probability P ( X > 3) and the upper bound given by the Markov inequality. Is the upper bound very tight? ( b ) Calculate both the exact probability P (1 ≤ X ≤ 5 / 3) and the upper bound given by the Chebyshev inequality. Is the upper bound very tight? Due:October 18, at 4 p.m. 2...
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