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# 276 Class Note for IE 58100 with Professor Schmeiser at Purdue

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This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 17 views.

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Date Created: 02/06/15

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 b c d e f g F The BoxMuller method of generating normal random variates is an example of composition 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 tx2 f x for every x 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 F Consider generating an ntrial 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 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 M1 is obtained depends upon the value of r 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 T When using the method of maximum likelihood to t a distribution to data x1 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 039 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 and 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 Schmeiser The densityfunction ofX isfxblb0 Sx Sb Therefore the likelihood function is n L03 11fxb il 1b310 SmaXx1x2 xnS b 1b310 s 83 Sb Then L b is maximized at the smallest value of b that is not less than 83 Therefore bA 83 Page 2 of4 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 quot and that its density function is fXx n l xquot71 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 1 Generate a random number u 1 2 Generate a random number u 2 3 Setx u 1 4 Sety cuz 5 Ify gtfxgoto Stepilt l gti 6 Retumx 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 of 4 Test 2 Spring 2000

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