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# Simulation Design And Analysis IE 58100

Purdue

GPA 3.98

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This 4 page Class Notes was uploaded by Dr. Cristian Lueilwitz on Saturday September 19, 2015. The Class Notes belongs to IE 58100 at Purdue University taught by Bruce Schmeiser in Fall. Since its upload, it has received 80 views. For similar materials see /class/208014/ie-58100-purdue-university in Industrial Engineering at Purdue University.

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Date Created: 09/19/15

IE 581 7 Introduction to Stochastic Simulation Name i lt KEY gt i One page of notes from and back Closed book 50 minutes 1 True or false If you wish write an explanation of your thinking a F The BoxMuller method of generating normal random variates is an example of composition b T In the acceptancerejection method of generating a random variate x from a speci ed density function f the inequality J txdx 2 1 must hold Here tx denotes the userchosen majorizing function owhich must satisfy tx 2 f x for every x c F When using the inverse cdf transformation to generate the next arrival time for a nonhomogeneous Poisson process the random variate is the time before the next arrival not the a1rival time itself d F Consider generating an 7 trial binomial random variate by generating 71 random numbers and counting the number of them that are less than p the probability of success This is an example of composition e F Consider using the threestep algorithm to generate a random vector x1x2 xr In Step 2 the random vector u1u2 ur is obtained The distribution from which u 1 is obtained depends upon the value of r f T When using the method of moments to t a distribution to data x1 x2 xquot the user must rst choose a family of distributions g T When using the method of maximum likelihood to t a distribution to data x 1 x2 xquot the user must rst choose a family of distributions h F To compute the empirical cdf of a data set x 1 x2 xquot the user must rst choose a number of cells i T Bootstrapping refers to drawing with replacement observations from a real world sample 139 F Your instructor is a strong advocate of goodnessof t testing to determine whether a tted distribution is an adequate simulation input model Schmeiser Page 1 of 4 Test 2 Spring 2000 IE 581 7 Introduction to Stochastic Simulation Name i lt KEY gt i 2 Consider the family of uniform distributions over the interval 0 b where the upperbound parameter b is unknown Suppose that you have realworld data x1 55 x2 72 an x3 83 a Estimate b using the method of moments BecauseX U 0 b the mean is EX b 2 The sample mean is X 55 72 83 3 7 Setting the distribution mean equal to the sample mean yields b 2 7 Therefore the method of moments yields the estimate 6 l4lt b Estimate b using maximum likelihood The densityfunction ofX isfxblb0 Sx Sb Therefore the likelihood function is n L03 H fxb il lb3I0SmaXx1x2 xnSb 1b310s83 Sb ThenL b is maximized at the smallest value of b that is not less than 83 Therefore bA 83 Schmeiser Page 2 of 4 Test 2 Spring 2000 IE 581 7 Introduction to Stochastic Simulation Name i lt KEY gt i 3 Consider a random variable X that takes values between zero and one Suppose that over this interval its cumulative distribution function is F Xx PX Sx l l x and that its density function is fXx n l x 1 Consider using acceptancerejection to generate a random variate x when the parameter value is n 3 We will use the uniform distribution with density function rx l on the interval 0 l and zero elsewhere a Sketch the density function f including labeling and scaling both axes Remember that the density is de ned for every real number x The density function is fXx 31 x 2 for 0 S x S l and zero elsewhere Sketch two perpendicular axes Label the horizontal axis x and indicate the points zero and one Indicate that the axis goes to both in nities Label the vertical axis f X x and scale it with zero and for example one The density is falls continuously from 3 atx 0 to 0 at x l The value at x 05 for example is 075 b The majorizing function tx will be a constant say 0 over the 0 l interval What is the value of c The majorizing function is tx c rx c l c for 0 Sx S l and zero elsewhere For all x in 0 lfx S tx Therefore 0 maxx fx 39 c In the following logic s Step 5 ll in the blank Generate a random number u 1 Generate a random number u 2 Set x u 1 Set y cuz Ify gtfx goto Stepilt l gti Return x 99WNH d In the logic of Part c what is the expected number of times that Step 5 is executed in order to return one random variate x Expected number of acceptancerejection trials is c 39 Schmeiser Page 3 of 4 Test 2 Spring 2000 IE 581 7 Introduction to Stochastic Simulation Name i lt KEY gt i 4 Consider the nonhomogeneous Poisson process having piecewiseconstant rate function Mt 5 when 0 S t S 10 Mt 7 when 10 lt t S 20 and zero elsewhere a Suppose that the simulation begins at time t0 As a function of I what is the expected number of arrivals The expected number of trials in the time interval 0 t is t At M10511 St for 0 St S 10 507t 10 for 10ltt S20 120 for 20 lt t lt oo b Consider generating arrivals from this process using the method of thinning from a homogeneous Poisson process with rate 1 At time t 15 what is the probability of rejecting a potential arrival The probablility ofrejection is 1 M15 1 1 71 e 5 Suppose you rerun a simulation experiment keeping everything the same except the randomnumber seeds Which are constants which are random and which are unde ned Circle one a 9 constante random unde ned b S constant randome unde ned c EX 0 constant random unde nedlt Schmeiser Page 4 of4 Test 2 Spring 2000

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