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# 08SimHw4 - IEOR 4404 Simulation Prof Mariana...

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IEOR 4404 Assignment #4 Simulation October 1, 2008 Prof. Mariana Olvera-Cravioto Page 1 of 2 Assignment #4 – due October 7th, 2008 1. Give inverse-transform, acceptance-rejection, and composition algorithms for generating from each of the following densities. Discuss which algorithm is preferable for each density. (First plot the densities). (a) f ( x ) = ( 3 x 2 2 , if - 1 x 1 , 0 , otherwise. (b) For 0 < a < 1 / 2, f ( x ) = x a (1 - a ) , if 0 x a, 1 1 - a , if a x (1 - a ) , 1 - x a (1 - a ) , if 1 - a x 1 , 0 , otherwise. 2. Using the composition method, give algorithms for generating random variables from the following distributions. (a) F ( x ) = ( x + x 3 + x 5 ) / 3, for 0 x 1 (b) F ( x ) = ( 1 - e - 2 x +2 x 3 , if 0 < x < 1 3 - e - 2 x 3 , if 1 < x < (c) F ( x ) = n i =1 α i x i , for 0 x 1, where α i 0, n i =1 α i = 1. 3. A pair of fair dice are to be continually rolled until all the possible outcomes 2 , 3 , . . . , 12 have occurred at least once. Develop a simulation study to estimate the expected number of dice rolls that are needed.

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08SimHw4 - IEOR 4404 Simulation Prof Mariana...

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