hw4_sol - EE 351K Probability, Statistics, and Random...

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EE 351K Probability, Statistics, and Random Processes SPRING 2009 Instructor: Shakkottai/Vishwanath shakkott, sriram@ece.utexas.edu Homework 4 - Solutions 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 50 dollars for being zero to two minutes behind, 30 dollars for being two to five minutes behind, 10 dollars for being 5 to 10 minutes behind, and nothing otherwise. Given that the difference between his time and the winner’s time is uniformly distributed between 0 and 15 minutes, find the mean and variance of the reward of the runner-up. Solution : Let X be the reward. The probability that X = 50 is 2 15 , the probability that X = 30 is 3 15 , and the probability that X = 10 is 5 15 . Therefore E [ X ] = 2 15 · 50 + 3 15 · 30 + 5 15 · 10 = 16 , and var ( X ) = E [ X 2 ] - ( E [ X ]) 2 = 2 15 · 50 2 + 3 15 · 30 2 + 5 15 · 10 2 - (16) 2 = 290 . 6667 . Problem 2 Let X be a random variable with PDF f X ( x ) = ± 3 x 2 7 if 1 < x 2 0 otherwise. and let Y = X 2 . Calculate E [ Y ] and var ( Y ) . Solution: We have E [ Y ] = E [ X 2 ] = Z 2 1 x 2 f X ( x ) dx = Z 2 1 3 x 4 7 dx = 3 5 x 5 7 ² ² ² 2 1 = 3 5 ( 32 7 - 1 7 ) = 93 35 = 2 . 6571 To obtain the variance of Y , we first calculate E [ Y 2 ] . We have, after straightforward calculation, E [ Y 2 ] = E [ X 4 ] = Z 2 1 x 4 f X ( x ) dx = Z 2 1 3 x 6
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hw4_sol - EE 351K Probability, Statistics, and Random...

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