Lecture9 - Lecture 9 Mariana Olvera-Cravioto Columbia...

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Lecture 9 Mariana Olvera-Cravioto Columbia University molvera@ieor.columbia.edu February 15th, 2012 IEOR 4404, Simulation Lecture 9 1/17
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Random permutations I Suppose we want to generate a random permutation of the elements 1 , 2 ,...,n , in such a way that each of the n ! possible permutations is equally likely. I Idea: Suppose we have balls numbered 1 , 2 ,...,n and we line them up in ascending order. We then pick randomly a number from { 1 , 2 ,...,n } , say K 1 , pick the corresponding ball and place it in position 1; once this is done we move the remaining balls to the left (preserving their relative order). We then pick a random number from { 1 , 2 ,...,n - 1 } , say K 2 , pick the ball in position K 2 and place it in position 2; once this is done we move the remaining balls to the left; and so on. 1 2 6 5 4 3 1 2 6 5 4 3 1 2 3 4 5 6 K = 3 1 1 2 6 5 4 3 1 2 6 5 4 3 1 2 3 4 5 6 K = 4 2 1 2 6 5 4 3 1 2 6 5 4 3 1 2 3 4 5 6 K = 1 3 IEOR 4404, Simulation Lecture 9 2/17
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Algorithm: 1. Set i = 0 , K i = (1 , 2 , 3 ,...,n ) and let Q be an empty vector. 2. Generate a Uniform number J i in { 1 , 2 ,...,n - i } 3. Add the J th i element of K i to the vector Q . 4. Let K i +1 be the vector obtained by removing the J th i element of K i . 5. If i < n , reset i = i + 1 and go to Step 2. Otherwise, return Q . IEOR 4404, Simulation Lecture 9 3/17
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Poisson Process I Suppose “events” occur at random time points and let N ( t ) = number of events in [0 ,t ] I Definition: These events constitute a Poisson process with rate λ , λ > 0 , if I N (0) = 0 I The numbers of events occurring in disjoint time intervals are independent. I The distribution of the number of events that occur in a given interval depends only on the length of the interval and not on its location. I lim h 0 P ( N ( h ) = 1) h = λ I lim h 0 P ( N ( h ) 2) h = 0 IEOR 4404, Simulation Lecture 9 4/17
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Properties of the Poisson process Result: The number of events occurring in an interval of length t is a Poisson r.v. with mean λt . Sketch of proof: IEOR 4404, Simulation Lecture 9 5/17
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Properties of the Poisson process Result: The number of events occurring in an interval of length t is a Poisson r.v. with mean λt . Sketch of proof:
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Lecture9 - Lecture 9 Mariana Olvera-Cravioto Columbia...

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