IE 581 Introduction to Stochastic Simulation

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1 1. List criteria for choosing the majorizing density r (x) when creating an acceptance/rejection random-variate generator for a specified density function f (x). 2. Suppose the rate function of a nonhomogeneous process is λ(t) = 3t 2. If we are currently at time 45.7 minutes, what is the inverse transformation to generate the next time t i? 3. Arrivals to a system are often modeled as a homogeneous Poisson process; that is, the times between arrivals are independent and exponential. An active research topic in queueing theory is to study the effect of autocorrelated times between arrivals on system behavior. How could you create a sequence of interarrival times that are each exponential (with mean µ a ) and with autocorrelation 0.3? Be specific. (A programmer who knows nothing of the theory should be able to code your logic.) 4. Random-variate generation via composition. Let X denote the processing time for a particular drilling operation. There are three types of parts: A, B, and C. Thirty percent are type A, 65 percent at type B, and 5 percent are type C. Drill time for a type A part is exponential with mean 3.2 minutes. Drill time for a type B part is uniformly distributed between 1.2 and 2.0 minutes. Drill time for a type C part is deterministically 0.5 minute. Parts arrive randomly and independently to the drill. (a) What is the cumulative distribution function of overall drill time? (b) Write composition logic to generate a random drill time. Be specific. 5. True or false. (a) The number of iterations for the acceptance/rejection method is always geometric. (b) The expected number of iterations for the acceptance/rejection method depends only upon the choice of the majorizing function cr(x). (c) The Box-Muller algorithm generates normal random variates via the inverse transformation. (d) Composition is applicable for generating either continuous or discrete random variables. (e) The observed average x of a sample uniformly distributed over the interval (a, b) is (a +b)/2. (f) Suppose n independent observations are grouped into k macroreplications of m microreplication each. The only advantage of having a large value of k is that the standard error will be better estimated. (g) Suppose n independent observations are grouped into k macroreplications of m microreplication each. The only advantage of having a large value of m is to that the standard error will be better estimated. 6. We used the multivariate normal distribution in the first step of the three-step algorithm for general random vectors. If we replaced this distribution with some other multivariate distribution, how (if it is possible at all) would we change the other algorithm steps? Schmeiser Page 1 of 8 Some previous Test-2 questions.

2 (a) T F Thinning is a way to generate realizations of a specified Poisson process. (b) T F A Poisson process has Poisson times between events. (c) T F The Box-Muller method uses two uniform random numbers to generate two independent standard normal random variates. (d) T F Maximum-likelihood estimation and the method of moments have the same goal: given data, estimate distribution parameters. (e) T F The likelihood of x 1 =.3 and x 2 = 1.3 is L = In our notes the four general methods for generating random variates are categorized as (1) inverse transformation, (2) composition, (3) acceptance/rejection, and (4) special properties. (a) Suppose the distribution from which we want to generate is given. Which of these four methods is then uniquely defined? (b) Law and Kelton discuss a fifth method: convolution. In convolution, a random variate is generated as the sum of i.i.d. simpler random variates. For example, a binomial random variable X might be generated by summing independent random variables X 1, X 2 through X n, where P(X i = 1) = p and P(X i = 0) = 1 p. Convolution is an important special case, but it is a special case of one of the four general methods. Which one? (c) Write logic (any sort of unambiguous pseudo code) to generate X 1 in Part (b). You may use any method. You may use the random-number generator u We studied a three-step procedure for generating random vectors (X 1, X 2,..., Xr ) with desired marginal distributions F Xi for i = 1,2,...,r and desired correlation matrix R. 1. Generate (z 1, z 2,..., zr ) from the multivariate normal distribution with zero means, unit variances, and correlation matrix R. 2. Compute (u 1, u 2,..., ur ) = (Φ(z 1 ),Φ(z 2 ),...,Φ(z r )), where Φ is the standard normal cumulative distribution function. 3. Compute (X 1, X 2,..., Xr ) using X i = F 1 Xi (u i ) for i = 1,2,...,r. (a) What is the role of the Cholesky decomposition in this algorithm? (b) What is the distribution of the random variate u 1 in this algorithm? (c) Do the random vectors generated from this algorithm have the specified marginal distributions? (d) Do the random vectors generated from this algorithm have the specified correlation structure? Schmeiser Page 2 of 8 Some previous Test-2 questions.

3 (a) T F In random-variate generation, convolution is a special case of the method of composition. (b) T F The inverse transformation for a U (3,10) distribution is x = 3 + 7u, where u is a random number. (Remark: U (a,b) is the continuous uniform distribution on the interval (a,b).) (c) T F The Cholesky decomposition C C t =Σ of the covariance matrix Σ is analogous to σ σ =σ 2 of the variance σ 2. (d) T F The inverse transformation of the normal distribution must be done numerically. That is, it is not closed form. (e) T F If the standard error of the point estimator is not being estimated, the optimal number of macro-replications is k = 1. (f) T F The standardized-moment plane (β 1, β 2 ) is useful for suggesting which family of distributions might be used to model an input random variable. (g) T F In the acceptance/rejection method to generate a random variate from a density function f using a majorizing density function r, the expected number of iterations is r (x) /f(x) dx. (h) T F The software package PRIME allows the user to fit one of several well-known distribution families to a set of real-world data. (i) T F An empirical cumulative distribution function is random. (j) T F A distribution with a long left tail has a negative skewness. (k) T F If the Poisson rate function is λ(t) = 2t, time between events tends to decrease as the simulation progresses. 2. The standardized-moments plane (α 3 2, α 4 ) = (β 1, β 2 ) is concerned with third and fourth moments. The first and second standardized moments are not interesting because they have the same value regardless of distribution. If E(X) =µand var(x) =σ 2, then (a) α 1 = E X µ = σ (b) α 2 = E X µ σ 2 = Schmeiser Page 3 of 8 Some previous Test-2 questions.

4 3. Random variates from the density function f X (x) = ln x, with range 0 x 1, can be generated using this algorithm: (i) Generate y U(0, 1) (ii) Generate x U(0, y). (a) This algorithm is an example of (circle all that apply): (i) inverse transformation (ii) composition (iii) probability mixing (iv) convolution (v) acceptance/rejection (b) Write C code (or pseudo code) to generate one random variate. (c) Show that the algorithm is valid. 4. In a certain system, requests to read from a computer disk arrive from background processes as a Poisson process with constant rate λ 1 (t) = 20 requests per minute. Independent of the background requests, foreground requests to read from the computer disk arrive as a Poisson process with constant rate λ 2 (t) = 10 requests per minute. (a) Then the arrival process for all read requests are a Poisson process with constant rate λ(t) = 30 requests per minute. This is an example of (circle all that are true) (i) thinning (ii) merging (iii) inverse transformation (b) To generate such requests in a simulation, one could use this algorithm: (i) Generate the time until the next request as exponential with mean 2 seconds. (ii) If a random number u satisfies u 1/3, the request is foreground, otherwise it is background. Here the foreground and background processes are both being generated using (circle all that are true) (i) thinning (ii) merging (iii) inverse transformation (c) Find the expected number of background requests from 8:30am until noon. Schmeiser Page 4 of 8 Some previous Test-2 questions.

5 5. The course instructor argued in class that goodness-of-fit tests are not appropriate for validating input models. Reasons supporting his position include (circle all that apply) (i) such tests are for statistical significance, not practical significance. (ii) such tests are for practical significance, not statistical significance. (iii) with lots of real-world data, even good models are often rejected by such tests. (iv) the results of such tests are independent of the modeler s deadlines and budgets. 6. The exponential density with mean µ is f X (x) = e x/µ / µ if x is positive and zero elsewhere. Suppose that you have one observation, x 1 = 10.3 days, of the random variable X. (a) Estimate µ using maximum likelihood. (b) Estimate µ using the method of moments. 7. We studied a three-step procedure for generating random vectors X = (X 1, X 2,...,X r ) with specified marginal distributions F i for i = 1, 2,..., r and specified correlation matrix R. (a) What are the dimensions of the matrix R? (b) In many situations, either the correlation matrix or the covariance matrix can be used. But here the correlation matrix R is used rather than the covariance matrix Σ because Σ contains redundant information. What is this redundant information? Schmeiser Page 5 of 8 Some previous Test-2 questions.

6 (a) T F The purpose of common random numbers is to reduce the standard errors of Y A and or Y B, the sample means from two systems, A and B, that are being compared. (b) T F For any two random variables X and Y, cov[x,y ] = cov[x µ X,Y µ Y ], where cov denotes covariance and µ X and µ Y are the respective means. (c) T F If Z 1 and Z 2 are independent standard normal random variables and ρ is a constant satisfying ρ 1, then X =ρz 1 + [(1 ρ 2 ] 1/2 Z 2 is also standard normal. 2. Suppose you rerun a simulation experiment, keeping everything the same except the random-number seeds. Which are constants and which are random? (a) Ê[Y ] (b) Pˆ[A] (c) θ (d) s.e. ˆ [pˆ] 6. Consider an integral a b t (x) dx, where t is a given integrable function and a and b are constants. Rewrite so that the integral can be interpreted as an expected value. (a) T F The number of iterations for the acceptance/rejection method is always geometric. (c) T F Composition is applicable for generating either continuous or discrete random variables. (d) T F The sample average x of a sample uniformly distributed over the interval (a, b) is(a +b)/2. (j) T F Common random numbers, a variance-reduction method, is designed to reduce the variance of Y 1, Y 2, and their difference Y 1 Y 2, where Y i is the sample average from system i, i = 1,2. 2. Which are constants and which are random? (Here "random" means that changing the random-number seed changes the value.) (a) Vˆ(pˆ) (b) E[Vˆ (pˆ)] (c) θˆ 7. Fill in the blanks. (a) In this course, the purpose of a simulation experiment is to estimate _. Schmeiser Page 6 of 8 Some previous Test-2 questions.

7 (b) We denote the point estimator by. (c) The standard error is the standard deviation of. (d) An example of a point estimator is. (e) Variance-reduction methods are designed to reduce the variance of. (f) Output-analysis methods estimate. 8. Consider using acceptance-rejection to generate a random variate x from a density that is triangular over the interval (a,b). Assume that the majorizing function t (x) is a constant s over the interval (a,b) and zero elsewhere. (a) What is set of values of s for which the algorithm is valid? (b) What is the optimal value of s? 6. Discuss generating standard normal random variates using acceptance/rejection with a uniform majorizing function. (Recall: The standard normal density function is φ(z) = exp(.5z 2 ) / (2π) over, ).) Schmeiser Page 7 of 8 Some previous Test-2 questions.

8 (a) T F In maximum likelihood estimation, the "maximum" is over all possible real-world samples x 1,x 2,...,x n that might have occurred. (b) T F The Poisson distribution is often a good model for the time to repair a machine. (c) T F "Composition" is a synonym for "probability mixing". (d) T F To use the method of acceptance/rejection for generating random variates from a specified distribution, the distribution must be continuous and in one dimension. (e) T F The Box-Muller method for generating normal random variates is an example of acceptance/rejection. (f) T F A random variable X whose distribution has skewness measure α 3 = 2.3 must be continuous. (g) T F Consider the empirical cumulative distribution function obtained by plotting the points x i,i/(n +1) for i = 1,2,...,n. Here x 1 denotes the first observation collected. (h) T F The correlation of X and Y, ρ(x,y), has the same numerical value as the covariance of X and Y in the special case when then means of X and Y are zero. (i) T F The cumulative distribution function of a continuous random variable X is U (0,1), regardless of the distribution of X. (j) T F The approximation x = [u.135 (1 u).135 ]/.1975 is an approximation to the cumulative distribution function of the standard normal distribution. (k) T F A point process describes the behavior of a point estimator. (l) T F The rate function λ(t) of a nonhomogeneous Poisson process can be discontinuous. (l) T F "Thinning" is a method for generating random vectors with specified marginal distributions and specified correlations matrix. 4. Suppose that a simulation practitioner is determining the input model for a random variable X by interviewing an expert. Further suppose that X is the time for an electronic device to "wake up" when turned on. (a) You have the choice of modeling this response time as U (0,c) or exponential with a mean of µ. Choose one and argue for your choice. (b) The expert judges that half the time the device responds in less than 12.4 seconds. Determine the corresponding parameter value for your choice in Part (a). Schmeiser Page 8 of 8 Some previous Test-2 questions.