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6 chapter8 Random Variate

# 6 chapter8 Random Variate - Chapter 8 Random-Variate...

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Chapter 8 Random-Variate Generation Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

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2 Purpose & Overview ± Develop understanding of generating samples from a specified distribution as input to a simulation model. ± Illustrate some widely-used techniques for generating random variates. ² Inverse-transform technique ² Acceptance-rejection technique ² Special properties
3 Inverse-transform Technique ± The concept: ² For cdf function: r = F(x) ² Generate r from uniform (0,1) ² Find x: x = F -1 (r) r 1 x 1 r = F(x)

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4 Exponential Distribution [Inverse-transform] ± Exponential Distribution: ² Exponential cdf: ² To generate X 1 , X 2 , X 3 r = F(x) = 1 – e - ± x for x ² 0 X i = F -1 (R i ) = -(1/ ± ) ln(1-R i ) [Eq’n 8.3] Figure: Inverse- transform technique for e xp( ± = 1)
5 Exponential Distribution [Inverse-transform] ± Example: Generate 200 variates X i with distribution exp( ± = 1) ² Generate 200 Rs with U(0,1) and utilize eq’n 8.3, the histogram of Xs become: ² Check: Does the random variable X 1 have the desired distribution? ) ( )) ( ( ) ( 0 0 1 0 1 x F x F R P x X P = ± = ±

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6 Other Distributions [Inverse-transform] ± Examples of other distributions for which inverse cdf works are: ² Uniform distribution ² Weibull distribution ² Triangular distribution
7 Empirical Continuous Dist’n [Inverse-transform] ±

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• Spring '14
• Probability distribution, Probability theory, Exponential distribution, Cumulative distribution function, special properties

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