Chapter5 - Chapter 5 Statistical Models in Simulation...

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Chapter 5 Statistical Models in Simulation Banks, Carson, Nelson & Nicol Discrete-Event System Simulation ٢ Purpose & Overview ± The world the model-builder sees is probabilistic rather than deterministic. ² Some statistical model might well describe the variations. ± An appropriate model can be developed by sampling the phenomenon of interest: ² Select a known distribution through educated guesses ² Make estimate of the parameter(s) ² Test for goodness of fit ± In this chapter: ² Review several important probability distributions ² Present some typical application of these models
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٣ Review of Terminology and Concepts ± In this section, we will review the following concepts: ² Discrete random variables ² Continuous random variables ² Cumulative distribution function ² Expectation ٤ Discrete Random Variables [Probability Review] ± X is a discrete random variable if the number of possible values of X is finite, or countably infinite. ± Example: Consider jobs arriving at a job shop. ± Let X be the number of jobs arriving each week at a job shop. ± R x = possible values of X (range space of X ) = {0,1,2,…} ± p(x i ) = probability the random variable is x i = P(X = x i ) ² p(x i ), i = 1,2, … must satisfy: ² The collection of pairs [x i , p(x i )], i = 1,2,…, is called the probability distribution of X , and p(x i ) is called the probability mass function (pmf) of X . = = 1 1 ) ( 2. all for , 0 ) ( 1. i i i x p i x p
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٥ Continuous Random Variables [Probability Review] ± X is a continuous random variable if its range space R x is an interval or a collection of intervals. ± The probability that X lies in the interval [a,b] is given by: ± f(x) , denoted as the pdf of X , satisfies: ± Properties X R X R x x f dx x f R x x f X in not is if , 0 ) ( 3. 1 ) ( 2. in all for , 0 ) ( 1. = = = b a dx x f b X a P ) ( ) ( ) ( ) ( ) ( ) ( . 2 0 ) ( because , 0 ) ( 1. 0 0 0 b X a P b X a P b X a P b X a P dx x f x X P x x p p p p = = = = = = ٦ Continuous Random Variables [Probability Review] ± Example: Life of an inspection device is given by X , a continuous random variable with pdf: ² X has an exponential distribution with mean 2 years ² Probability that the device’s life is between 2 and 3 years is: = otherwise , 0 0 x , 2 1 ) ( 2 / x e x f 14 . 0 2 1 ) 3 2 ( 3 2 2 / = = dx e x P x
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٧ Cumulative Distribution Function [Probability Review] ± Cumulative Distribution Function (cdf) is denoted by F(x) , where F(x) = P(X <= x) ² If X is discrete, then ² If X is continuous, then ± Properties ± All probability question about X can be answered in terms of the cdf, e.g.: = x x i i x p x F all ) ( ) ( = x dt t f x F ) ( ) ( 0 ) ( lim 3. 1 ) ( lim 2. ) ( ) ( then , If function. ing nondecreas is 1. = = −∞ x F x F b F a F b a F x x p b a a F b F b X a P p p all for , ) ( ) ( ) ( = ٨ Cumulative Distribution Function [Probability Review] ± Example: An inspection device has cdf: ² The probability that the device lasts for less than 2 years: ² The probability that it lasts between 2 and 3 years: 2 / 0 2 / 1 2 1 ) ( x x t e dt e x F = = 632 . 0 1 ) 2 ( ) 0 ( ) 2 ( ) 2 0 ( 1 = = = = e F F F X P 145 . 0 ) 1 ( ) 1 ( ) 2 ( ) 3 ( ) 3 2 ( 1 ) 2 / 3 ( =
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Chapter5 - Chapter 5 Statistical Models in Simulation...

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