Lecture7

# Lecture7 - Lecture 7 Mariana Olvera-Cravioto Columbia...

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Unformatted text preview: Lecture 7 Mariana Olvera-Cravioto Columbia University [email protected] February 8th, 2012 IEOR 4404, Simulation Lecture 7 1/18 How to choose g ( x ) The majorizing function g ( x ) must satisfy two purposes: I We must be able to generate from density h ( x ) = g ( x ) /c easily. I Our choice of g ( x ) should make the probability of rejection small (choose g ( x ) as close to f ( x ) as possible) In our Beta(4,3) we could have chosen g ( x ) as follows: f(x) x x g(x) 3/5 1 f(x) x g(x) 3/5 1 2.074 h(x) IEOR 4404, Simulation Lecture 7 2/18 Acceptance-rejection method for discrete r.v. Suppose X is a discrete r.v. that takes values { x 1 ,x 2 ,x 3 ,... } and has PMF p ( x ) . Choose the majorizing function g ( x ) such that g ( x i ) ≥ p ( x i ) for all i = 1 , 2 , 3 ,... and set c = ∑ ∞ i =1 g ( x i ) . Define h ( x ) = g ( x ) /c . Algorithm: 1. Generate Y having PMF h ( x ) 2. Generate U ∼ Uniform[0,1], independent of Y 3. If U ≤ p ( Y ) /g ( Y ) , return X = Y . Otherwise, go back to step 1 and try again. IEOR 4404, Simulation Lecture 7 3/18 Pros and cons of the acceptance-rejection method Disadvantages: I Requires more than one Uniform[0,1] random variates (at least two) to generate X . I A bad choice of the majorizing function g ( x ) may lead to a big probability of rejection, which means more random numbers will be needed. I We cannot know in advance how many iterations of the algorithm we are going to need to generate X . Advantages: I The method works even for very complicated distributions provided we know their PDF, f ( x ) , or PMF, p ( x ) . I The method works particularly well when X has finite support, since we can choose g ( x ) to be a constant (maybe not the best choice, but certainly simple to simulate from). IEOR 4404, Simulation Lecture 7 4/18 Composition method I Suppose we want to generate a r.v. X whose CDF F can be written as F ( x ) = α 1 F 1 ( x ) + α 2 F 2 ( x ) + ··· = ∞ X i =1 α i F i ( x ) where...
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## This note was uploaded on 03/18/2012 for the course IEOR 4404 taught by Professor C during the Spring '10 term at Columbia.

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Lecture7 - Lecture 7 Mariana Olvera-Cravioto Columbia...

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