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# Hw4 - 351K EE Probability Statistics and Random Processes...

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351K EE Probability, Statistics, and Random Processes Instructor: Shakkottai/Vishwanath Homework 4 - Solutions SPRING 2008 shakkott, Problem 1 The runner-up in a road race is given a reward that depends on the difference between his time and the winner's time. He is given 20 dollars for being one minute behind, 10 dollars for being one to two minutes behind, 5 dollars for being 2 to 6 minutes behind, and nothing otherwise. Given that the difference between his time and the winner's time is uniformly distributed between 0 and 12 minutes, find the mean and variance of the reward of the runner-up. Solution : Let X be the reward. The probability that X = 20 is probability that X = 5 is 1 . Therefore 3 1 12 , the probability that X = 10 is 1 12 , and the E[X] = 0.083 20 + 0.083 10 + 0.333 5 = 4.167, and var(X) = E[X 2 ] - (E[X])2 = 0.083 202 + 0.083 102 + 0.333 52 - (4.167)2 = 32.636. Problem 2 Let X be a random variable with PDF fX (x) = and let Y = X 2 . Calculate E[Y ] and var(Y ). Solution: We have E[Y ] = E[X 2 ] = 2 2x/5 if 2 < x 3, 0 otherwise. 3 3 x2 fX (x) dx = 2 2 x4 2x3 dx = 5 5 4 3 3 2 = 2 81 16 ( - ) = 6.5 5 4 4 To obtain the variance of Y , we first calculate E[Y 2 ]. We have, after straightforward calculation, 3 E[Y 2 ] = E[X 4 ]

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Hw4 - 351K EE Probability Statistics and Random Processes...

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