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hw9-sol - AMS 311(Fall 2010 Joe Mitchell PROBABILITY THEORY...

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AMS 311 (Fall, 2010) Joe Mitchell PROBABILITY THEORY Homework Set # 9 – Solution Notes (1). (20 points) The AMS department receives, on average, three requests per day for students to sign into the major. We do not know the probability distribution for the number, X i , of students who sign into the AMS major on day i . Let X i be the number of students who sign into the AMS major on day i . We know that E ( X i ) = 3. (a). Let p be the probability that five or more students sign into the AMS major on Monday. Give the best guaranteed estimate you can for the probability p . (What inequality are you using?) (We think of Monday as “day i ”.) We use Markov’s inequality to get a bound on p : 0 p = P ( X i 5) E ( X i ) 5 = 3 5 = 0 . 6 (b). For the next three parts ((b), (c), (d)) assume that we also know that the variance, var ( X i ) , is 9. Try now to use an appropriate Chebyshev inequality to give an improved pair of bounds (upper and lower) on the probability p . Now we know that var ( X i ) = 9. So we apply the one-sided Chebyshev inequality, with μ = E ( X i ) = 3 and a = 2: 0 P ( X i 5) = P ( X i 3 + 2) 9 9 + 2 2 = 9 13 = 0 . 692 Note that, in fact, in this case the Chebyshev inequality gives a weaker bound than the Markov inequality did in (a). (This answers a question that was asked in lecture!) (You could have used a 2-sided Chebyshev, which would result in a weaker bound as follows: P ( X i 5) P ( X i 5 or X i 1) = P ( | X i - 3 | ≥ 2) 9 2 2 = 9 4 = 2 . 25. In fact, this is such a weak bound that it is useless, since 2 . 25 > 1.) (c). Give a Central Limit Theorem estimate for the probability q that more than 75 students sign into the AMS major during this month (December, which has 31 days, and we consider each day to be like any other day). Let X i be the number of students signing into the AMS major on December i , for i = 1 , . . . , 31. Let X = 31 i =1 X i denote the total number of students signing into AMS major in December. Then, E ( X ) = 31 · E ( X i ) = 31 · 3 = 93 and var ( X ) = 31 · var ( X i ) = 279 (using the fact that the X i ’s are independent). Then, using the Central Limit Theorem estimate, we get q = P ( X > 75) = P ( X - 93 279 > 75 - 93 279 ) P ( Z > - 1 . 0776) Φ(1 . 08) = . 8599 (d). Use an inequality to get the best bounds you can on the probability q estimated in part (c).
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