08SimHw3 - n and p . Suppose that we want to generate a...

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IEOR 4404 Assignment #3 Simulation September 23, 2008 Prof. Mariana Olvera-Cravioto Page 1 of 1 Assignment #3 – due September 30th, 2008 1. If x 0 = 5 and x n = 2 x n - 1 mod(150) find x 1 ,...,x 10 . 2. Use simulation to approximate the following integrals. Compare your estimate with the exact answer if known. (a) R 2 - 2 e x + x 2 dx (b) R 0 x (1 + x 2 ) - 2 dx (c) R 1 0 R 1 0 e ( x + y ) 2 dydx 3. For uniform (0,1) random variables U 1 ,U 2 ,... define N = min ( n : n X i =1 U i > 1 ) That is, N is equal to the number of random numbers that must be summed to exceed 1. (a) Estimate E [ N ] by generating 100 values of N . (b) Estimate E [ N ] by generating 1000 values of N . (c) Estimate E [ N ] by generating 10,000 values of N . (d) What do you think is the value of E [ N ]? 4. Use simulation to approximate cov( U,e U ), where U is uniform on (0,1). Compare your approximation with the exact answer. 5. Let X be a binomial random variable with parameters
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Unformatted text preview: n and p . Suppose that we want to generate a random variable Y whose probability mass function is the same as the conditional mass function of X given that X ≥ k , for some k ≤ n . Let α = P ( X ≥ k ) and suppose that the value of α has been computed. (a) Give the inverse transform method for generating Y . (b) Give a second method for generating Y . (c) For what values of α , small or large, would the algorithm in (b) be inefficient? 6. Give a method for generating a random variable having density function f ( x ) = e x / ( e-1) , ≤ x ≤ 1 7. Extra credit: If Z is a standard normal random variable, show that E [ | Z | ] = ± 2 π ² 1 / 2...
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This note was uploaded on 06/03/2011 for the course IEORIEORIE 4004 taught by Professor Simulation during the Spring '08 term at Columbia.

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