ps12_sol - Massachusetts Institute of Technology Department...

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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Solutions for Problem Set 12: Topic: Central Limit Theorem Due: No due date 1. It is not easy to calculate the CDF of the total weight and the desired probability, but an approximate answer can be quickly obtained using the central limit theorem. We want to calculate P ( S 102 > 200), where S 102 is the sum of the weights of 102 pretzels. The mean weight of a single pretzel is µ W = 2 . The variance of the weight of a single pretzel is 2 σ 2 = E [ W 2 ] − µ W W = ∞ w 2 f W ( w ) dw − 4 2 3 = w 2 ( w − 1) dw + w 2 (3 − w ) dw − 4 1 2 25 = 6 − 4 1 = 6 . Thus we have P ( S 102 > 200) = 1 − P ( S 102 ≤ 200) S 102 − 102 · 2 200 − 102 · 2 = 1 − P 102 · (1 / 6) ≤ 102 · (1 / 6) 1 − Φ − 4 ≈ √ 17 4 = 1 − 1 − Φ √ 17 = Φ(0 . 9701) ≈ . 8340 . 2. The probability that you will believe the fair coin to be biased is the probability that the fair coin will come up with more than 525 heads out of the 1000 tosses. Let S be the number of times the coin comes up heads, which is a binomial random variable, with parameters...
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This note was uploaded on 12/08/2010 for the course MATH 6.041 / 6. taught by Professor Muntherdahleh during the Spring '10 term at MIT.

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ps12_sol - Massachusetts Institute of Technology Department...

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