rec21 - (b Find(approximately the largest value of n such...

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Massachusetts Institute of Technology 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 21 November 23, 2010 1. Let X 1 ,...,X 10 be independent random variables, uniformly distributed over the unit interval [0,1]. (a) Estimate P ( X 1 + ··· + X 10 7) using the Markov inequality. (b) Repeat part (a) using the Chebyshev inequality. (c) Repeat part (a) using the central limit theorem. 2. Problem 10 in the textbook (page 290) A factory produces X n gadgets on day n , where the X n are independent and identically dis- tributed random variables, with mean 5 and variance 9. (a) Find an approximation to the probability that the total number of gadgets produced in 100 days is less than 440.
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Unformatted text preview: (b) Find (approximately) the largest value of n such that P ( X 1 + ··· + X n ≥ 200 + 5 n ) ≤ . 05 . (c) Let N be the ±rst day on which the total number of gadgets produced exceeds 1000. Cal-culate an approximation to the probability that N ≥ 220. 3. Let X 1 ,X 2 ,..., be independent Poisson random variables with mean and variance equal to 1. For any n > 0, let S n = ∑ n i =1 X i . (a) Show that S n is Poisson with mean and variance equal to n . Hint: Relate X 1 ,X 2 ,...,X n to a Poisson process with rate 1. (b) Show how the central limit theorem suggests the approximation n ! ≈ √ 2 πn p n e P n for large values of the positive integer n . Page 1 of 1...
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This note was uploaded on 12/02/2011 for the course ENGINEERIN EE302 taught by Professor Proasin during the Spring '11 term at South Carolina.

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