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Lecture6

# Lecture6 - Lecture 6 Mariana Olvera-Cravioto Columbia...

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Lecture 6 Mariana Olvera-Cravioto Columbia University [email protected] February 6th, 2012 IEOR 4404, Simulation Lecture 6 1/17

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Generating random variates Assume that we are able to generate iid Uniform[0,1] random variates. Method 1: Inverse Transform Suppose we want to generate a r.v. X such that I X is continuous with CDF F ( x ) I F ( x ) is continuous and strictly increasing when 0 < F ( x ) < 1 (it can be just nondecreasing for F ( x ) = 0 or F ( x ) = 1 ) I F ( x ) has an inverse F - 1 ( x ) . Algorithm: 1. Generate U Uniform[0,1] 2. Return X = F - 1 ( U ) IEOR 4404, Simulation Lecture 6 2/17
Inverse Transform To see that X , generated by the inverse transform method, has the correct distribution note that: IEOR 4404, Simulation Lecture 6 3/17

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Inverse Transform To see that X , generated by the inverse transform method, has the correct distribution note that: P ( X x ) = P ( F - 1 ( U ) x ) = P ( U F ( x )) Since P ( U u ) = 0 , if u < 0 , u, if 0 u 1 , 1 , if u > 1 , then P ( X x ) = P ( U F ( x )) = F ( x ) IEOR 4404, Simulation Lecture 6 3/17
Example We want to generate an exponential r.v. with mean β . IEOR 4404, Simulation Lecture 6 4/17

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Example We want to generate an exponential r.v. with mean β . We start by writing the CDF of X : F ( x ) = ( 1 - e - x/β , if x 0 0 , otherwise. To invert F ( x ) : y = 1 - e - x/β e - x/β = 1 - y - x β = log(1 - y ) x = - β log(1 - y ) Therefore, F - 1 ( x ) = - β log(1 - x ) . So if U Uniform[0,1], then X = - β log(1 - U ) Exponential (1 ) IEOR 4404, Simulation Lecture 6 4/17
Note: If U is uniformly distributed in [0,1], then 1 - U is too, so we could have used X = - β log U IEOR 4404, Simulation Lecture 6 5/17

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Inverse transform method I The inverse transform method can also be used to generate discrete random variables.
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Lecture6 - Lecture 6 Mariana Olvera-Cravioto Columbia...

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