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Unformatted text preview: IEOR E4703: Practice Midterm Exam With Solutions, Fall 2010. Professor Sigman. 1. X 1 and X 2 are two independent random variables distributed as: P ( X 1 = 0) = 0 . 30 , P ( X 1 = 1) = 0 . 50 , P ( X 1 = 2) = 0 . 20 and P ( X 2 = 1) = 0 . 40 , P ( X 2 = 3) = 0 . 60 (a) Give an algorithm for generating from X = X 1 X 2 that uses two uniform numbers U 1 , U 2 . SOLUTION: Use the discrete inverse transform method to produce, from U 1 and U 2 , X 1 and X 2 , then set X = X 1 X 2 : If U 1 < . 30 , set X 1 = 0; if 0 . 30 U 1 < . 80 , set X 1 = 1; if U 1 . 80 , set X 1 = 2. If U 2 < . 40 , set X 2 = 1; if U 2 . 40 , set X 2 = 3. Set X = X 1 X 2 . (b) Give an algorithm for generating from X = X 1 X 2 that uses only one uniform number U . SOLUTION: Derive explicitly, the distribution of X and use the inverse transform method on it. P ( X = 0) = 0 . 30 P ( X = 1) = P ( X 1 = 1) P ( X 2 = 1) = 0 . 20 P ( X = 2) = P ( X 1 = 2) P ( X 2 = 1) = 0 . 08 P ( X = 3) = P ( X 1 = 1) P ( X 2 = 3) = 0 . 30 P ( X = 6) = P ( X 1 = 2) P ( X 2 = 3) = 0 . 12 . If U < . 30 , set X = 0; if 0 . 30 U < . 50 , set X = 1; if 0 . 50 U < . 58 , set X = 2 if 0 . 58 U < . 88 , set X = 3; if U . 88, set X = 6. (c) Suppose that X has an exponential distribution at rate = 2; F ( x ) = P ( X x ) = 1 e 2 x . Suppose you wish to generate a rv Y that has the conditional distribution of X given that it falls in the interval (4 , 10). Show that the following algorithm works: i. Generatei....
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 Spring '07
 sigman

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