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# hw2_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 2 - Solution Problem 1 A new test has been developed to determine whether a given student is overstressed. This test is 80% accurate if the student is not overstressed, but only 70% accurate if the student is in fact over- stressed. It is known that 50% of all students are overstressed. Given that a particular student tests negative for stress, what is the probability that the test results are correct, and that this student is not overstressed? Solution : Let A be the event that the student is not overstressed, and let A c be the event that the student is in fact overstressed. Now let B be the event that the test results indicate that the student is not overstressed. The desired probability, P ( A | B ) , is found by Bayes’ rule: P ( A | B ) = P ( A ) P ( B | A ) P ( A ) P ( B | A ) + P ( A c ) P ( B | A c ) = 0 . 5 · 0 . 8 0 . 5 · 0 . 8 + 0 . 5 · 0 . 3 0 . 7273 . Problem 2 A parking lot consists of a single row containing n parking spaces ( n 2) . Mary arrives when all spaces are free. Tom is the next person to arrive. Each person makes an equally likely choice among all available spaces at the time of arrival. Describe the sample space. Obtain P ( A ) , the probability the parking spaces selected by Mary and Tom are at most 1 space apart. Solution 1: For convenience, we will number each of the parking spaces. parking space in one end is numbered 1, and sequentially numbered until numbering the other end n. Mary can choose any of the n parking spaces. She has a probability of 1 /n of selecting any particular space. Tom can choose any of the remaining n - 1 spaces and has a probability of 1 / ( n - 1) of choosing any particular space (other than the one Mary chose). Let M i be the event that Mary choose the parking space i , 1 i n , and T i be the event that Tom choose the parking space i , 1 i n . Also, let A be the event that the parking spaces selected by Mary and Tom are 1 space apart (at most 1 space apart). Note that Tom has one parking space next to Mary if Mary picks

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hw2_sol - EE 351K Probability Statistics and Random...

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