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Unformatted text preview: SIEO 3600 (IEOR Majors) Solution of Assignment #10 Introduction to Probability and Statistics April 13, 2010 Solution of Assignment #10 ( Central limit theorem, maximum likelihood estimator ) 1. Charity University announces a fund raising target of $10 millions. It sends out n letters to its alumni. From past experience, there is a probability of 0.6 that each alumni will not respond at all. Given that an alumni does respond (with probability 0.4), the amount of contribution is a random variable with mean $ and standard deviation $ . (a) What is the mean and variance of the amount of contribution from each letter sent? (b) We can reasonably assume that each letter recipient will behave independently with the above characteristic. What is the minimum number of letters to send so that Charity University will hit its target with at least 90% probability? Hint: Think about the fire-hydrant example discussed in class. Solution: (a) Denote X i as the amount of contribution of the i th alumni. We have E [ X ] = . 4 + 0 . 6 = 0 . 4 , E [ X 2 ] = ( 2 + 2 ) . 4 + 0 . 6 = 0 . 4( 2 + 2 ) , var( X ) = E [ X 2 ]- ( E [ X ]) 2 = 0 . 24 2 + 0 . 4 2 . (b) We want to find the smallest n such that . 9 P ( n X i =1 X i 10 million) = P n i =1 X i- n E [ X ] p n var( X ) 10 million- n E [ X ] p n var( X ) ! P N (0 , 1) 10 million- n E [ X ] p n var( X ) ! (by CLT) = 1- 10 million- n E [ X ] p n var( X ) ! , which implies that 10 million- n E [ X ] p n var( X ) (0 . 1) = z . 9 = 1 . 29(from the normal table) We can solve for n from the above inequality. 2. A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the probability that (a) you are winning after 34 bets; (b) you are winning after 1,000 bets; (c) you are winning after 100,000 bets. 2 SIEO 3600, Solution of Assignment #10 Solution: Let X i be the payoff in the i th bet, then X i = 35 , p = 1 / 38 ,- 1 , p = 37 / 38 ....
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This note was uploaded on 10/03/2011 for the course SIEO W3600 taught by Professor Yunanliu during the Spring '10 term at Columbia.
- Spring '10