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Unformatted text preview: EE 351K PROBABILITY & RANDOM PROCESSES FALL 2011 Instructor: Sujay Sanghavi sanghavi@mail.utexas.edu Homework 9 Solution Problem 1 The parameter of an exponential random variable has to be estimated from one sample. What is the ML estimator? Is it unbiased? Sol : Let be the parameter of exponential distribution. Then the likelihood of single sample X = x is f X ( x ; ) = e x . We want to maximize this probability with respect to , then from d d e x = e x xe x = 0 , we have ML = arg max p X ( x ; ) = 1 /x. So the estimator is ML = 1 X . By definition, it is easy to get E [ ] = E [ 1 X ] = 1 x e x dx. Also note that 1 x e x dx = ln x x 1! + 2 x 2 2 2! 3 x 3 3 3! + . So E [ ] = , which implies the estimator is biased. Problem 2 Alice models the time that she spends each week on homework as an exponentially distributed random variable with unknown parameter . Homework times in different weeks are independent. After spending 10, 14, 18, 8, and 20 hours in the first 5 weeks of the semester, what is her ML estimator of ? Sol : The likelihood can be expressed as f X ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ; ) = e 10 e 14 e 18 e 8 e 20 = 5 e 70 , so ML = arg max p X ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ; ) = 1 14 . Problem 3 X is known to be a uniform random variable, with range [ a,a ] . However, the parameter a is unknown, and has to be estimated from n samples. (a) What is the ML estimator?...
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This note was uploaded on 02/20/2012 for the course EE 351k taught by Professor Bard during the Spring '07 term at University of Texas at Austin.
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