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# Class Note for EMSE 273 with Professor Dorp at GW (6)

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This 20 page Class Notes was uploaded by an elite notetaker on Saturday February 7, 2015. The Class Notes belongs to a course at George Washington University taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/07/15

EMSE 273 Discrete System Simulation MONTE CARLO SIMULATIONINTEGRATION F INPUT UNCERTAINTY CaCuate E Sample X1 Y1 z1 01 i I Calculate 39 Sample X2Y2Z2 gt 02 Calculate I Sample X3Y3Z3 gt 03 I I SOLLSLLViS ETC it MODEL m I I III III 39 gt I m FX Y Z Inn 5 a a Um I an I m I m I m m mm m II m an Inn m I Lecture Notes by Instructor Dr J Rene van Dorp Page 1 EMSE 273 Discrete System Simulation RANDOM VARIATE GENERATION Let U be a Uniform Random Variable on 01 with density function fu and cumulative distribution function Fu O ult0 fu 1 032131 0 O Fu PrU S u Lecture Notes by Instructor Dr J Rene van Dorp Page 2 EMSE 273 Discrete System Simulation DISCRETE RANDOM NUMBER GENERATION A u Pra g u g b A ba H 0 SUPPOSE YOU WANT TO SAMPLE A REALIZATION X FROM A DISCRETE RANDOM VARIABLE X X6x19 9xn9PrX xipi PrX ltxz39 ippPrX 3x1 11 11 Lecture Notes by Instructor Dr J Rene van Dorp Page 3 EMSE 273 Discrete System Simulation STEP 1 Generate a realization u of the random variable U STEP 2 x1 0 S u lt p1 X2 1913 u lt 191 192 i l l xltxl pj ult p 11 11 n l xn p S u lt1 t H Lecture Notes by Instructor Dr J Rene van Dorp Page 4 EMSE 273 Discrete System Simulation INTERMEZZO VERTICAL LOOKUP FUNCTION Let Xy1 y29m9ynthen yl yl Sulty2 L00kupu y yn l yn lgultyn y u 2 yn EXCEL has two functions which are based around the LookUp function VLOOKUP and HLOOKUP If your vector is a column vector you use VLOOKUP If your vector is a row vector you use HLOOKUP Lecture Notes by Instructor Dr J Rene van Dorp Page 5 EMSE 273 Discrete System Simulation EXAMPLE SAMPLER FOR DISCRETE DEMAND DISTRIBUTION 030 100 Input Distribution 020 150 050 040 030 Demand 020 010 000 100 150 200 250 300 Lecture Notes by Instructor Dr J Rene van Dorp Page 6 EMSE 273 Discrete System Simulation INPUT DISTRIBUTION PrXlt PrXlt Outcome Outcome Outcomes Probabilities 1 000 030 100 030 2 030 050 150 020 3 050 080 200 030 4 080 095 250 015 5 095 100 300 005 Random U 055 Realization DEMAND 200 Using DATATABLE and RANDOM NUMBER GENERATOR in EXCEL we can create a Sample of the Discrete Demand Distribution Note RANDOM NUMBER GENERATOR s typically generate realization of a uniform random variable on 01 Lecture Notes by Instructor Dr J Rene van Dorp Page 7 EMSE 273 Discrete System Simulation CONTINUOUS RANDOM NUMBER GENERATION Suppose X is a CONTINUOUS random variable with cumulative distribution function F PrX S x Fx A random Variable Z would also have cumulative distribution function F if PrZ S Z FZ Lecture Notes by Instructor Dr J Rene van Dorp Page 8 EMSE 273 Discrete System Simulation yFltxgt 100 150 200 250 300 X CONCLUSION CDF is Continuous Strictly Increasing Function Therefore Fx has a well defined inverse function F391yx Lecture Notes by Instructor Dr J Rene van Dorp Page 9 EMSE 273 Discrete System Simulation THEOREM Let X be a continuous random variable with cdf Fx Let U be a uniform random variable on 01 Let Z be the random variable such that Z F391U v Z is a continuous random variable with cdf Fz HOMEWORK Proof the theorem above Lecture Notes by Instructor Dr J Rene van Dorp Page 10 EMSE 273 Discrete System Simulation SAMPLING ALGORITHM 1 100 A 090 080 STEP 1 Sample 070 Realization u from 060 7 Uniform Random 050 Variable U 040 39 030 i 020 l 010 39 0 V 000 i i 150 200 I 250 300 I I STEP 2 Calculate XF1u realization X F391 u from Random Variable Lecture Notes by Instructor Dr J Rene van Dorp Page 11 EMSE 273 Discrete System Simulation EXAMPLE Sampling from a Normal Distribution with Mean 200 Standard Deviation 30 Mean 200 Standard Deviation 30 Random U 055 Realization DEMAND 20376 Using DATATABLE and RANDOM NUMBER GENERATOR in EXCEL we can create a Sample of the Continuous Distribution Note RANDOM NUMBER GENERATOR s typically generate realizations of a uniform random variable on 01 Lecture Notes by Instructor Dr J Rene van Dorp Page 12 EMSE 273 Discrete System Simulation INPUT DISTRIBUTION NORMAL20030 10000 8000 6000 4000 2000 000 CDF Input Distribution 100 120 140 160 180 200 220 240 260 280 Lecture Notes by Instructor Dr J Rene van Dorp 13 Page EMSE 273 Discrete System Simulation EMPIRICAL DISCRETE PDF ESTIMATION The Empirical Probability Mass function is constructed using relative frequencies of events of occurences of particular outcome Relative Frequency 2 Total number of occurrences Lecture Notes by Instructor Dr J Rene van Dorp Page 14 EMSE 273 Discrete System Simulation Estimation of Discrete Empirical Probability Mass Function Accidents Occurrences PrAccidents O No N0M 1 N1 MM 2 N2 N2M 3 N3 N3M 4 N4 MM 4 Total M ZNI39 10 Lecture Notes by Instructor Dr J Rene van Dorp Page 15 EMSE 273 Discrete System Simulation EMPIRICAL PDF DEMAND INPUT DISTRIBUTION Sampled Distribution Sample Size 100 05 04 03 02 01 0 I I I I 100 150 200 250 300 CONCLUSION Sample Size of 100 is not LARGE ENOUGH Lecture Notes by Instructor Dr J Rene van Dorp Page 16 EMSE 273 Discrete System Simulation ESTIMATION OF EMPIRICAL CONTINUOUS CDF Y Failure Time of Machine 1Given data yr i 1n 2 Order data such that y1lt ya lt 39quotlt yn 1lt 301 3 Set 1 Fy1 PIG S y1 Z 2 11 1 Fy2 PTO S y2 Z Fyn71 PrY 3 320171 2 T n Fyn PTO S 30192 1 4 Plot the points y1Fy17 ynFyn in a graph 5 Connect these points by a straight line Lecture Notes by Instructor Dr J Rene van Dorp Page 17 EMSE 273 Discrete System Simulation SAMPLED EMPIRICAL CDF 10000 EMPIRICAL CDF Input Distribution Sample Size 100 8000 6000 4000 2000 000 100 150 200 250 300 Sample Size of 100 is not LARGE ENOUGH Lecture Notes by Instructor Dr J Rene van Dorp 18 Page EMSE 273 Discrete System Simulation ESTIMATION OF MPOINT EMPIRICAL PROBABILITY MASS FUNCTION or HISTOGRAM APPROACH Develop a DISCRETE APPROXIMATION of CONTINUOUS PDF by assigning probability mass on the interval ab to the midpoint of this interval ie ab2 NOTE PrY 6 ab FbFb Mpoint approximation method of PDF 1Given data yr i 1n 2 Order data such that yo lt y2 lt lt yn 1 lt Mn 3 Calculate yon ya Zj y1J m j1m Lecture Notes by Instructor Dr J Rene van Dorp Page 19 EMSE 273 Discrete System Simulation 4 For every Zj determine yU such that yo lt Z lt yz391 5Setforj01 m i FZ I l 6Setforj1 m ZJ1ZJ PM T Fltzjgt FltzJ1gtJ STEPS 4 5 and 6 can be executed in EXCEL using the FREQUENCY FUNCTION Lecture Notes by Instructor Dr J Rene van Dorp Page 20

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