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# IE306Lec7 - IE306 SYSTEMS SIMULATION Ali Rza Kaylan...

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1 IE306 IE306 SYSTEMS SIMULATION SYSTEMS SIMULATION Ali Rıza Kaylan [email protected]

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2 Random Variate Generation Random Variate Generation Inverse Transformation Technique Inverse Transformation Technique Composition Technique Composition Technique Convolution Technique Convolution Technique Acceptance Acceptance - - Rejection Technique Rejection Technique Poisson Variate Generation Poisson Variate Generation Normal Variate Generation Normal Variate Generation LECTURE 7 LECTURE 7 OUTLINE OUTLINE
3 RANDOM VARIATE GENERATION RANDOM VARIATES X~F(x) {X1,X2,X3,...} RANDOM NUMBERS U~IIDUniform(0,1) {U1,U2,U3,...} f ( u ) = 1 0 u 1 0 Otherwise F ( u ) = 0 u < 0 u 0 u < 1 1 1 u

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4 RANDOM VARIATE GENERATION X ~ Uniform (2,6) X = 2 + (6 2) U x f(u) f(u) 0 2 6 X=5 0.25 u 0 1 U=0.75 EXAMPLE:
5 RANDOM VARIATE GENERATION F ( x ) = 0 x < 2 x - 2 4 2 x < 6 1 6 x x F(x) 0 2 6 X=5 1 0.75

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6 INVERSE TRANSFORMATION TECHNIQUE 1. Generate a random number U~Uniform(0,1) 2. Set Example: X = F 1 U ( ) P ( X x ) = P ( F 1 ( U ) x ) = P ( U F ( x )) = F ( x ) X ~ Uniform (2,6) 4 2 - X = U U X 4 2 + =
7 INVERSE TRANSFORMATION TECHNIQUE Example: X ~ Exponential ( β ) F ( x ) = 1 e ( x / β ) ,0 x U = 1 e ( X / β ) X = - β ln(1 U )

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