ISYE 6644 OH Exam 1 Practice Questions - Spring 2025
Complete Study Guide with Verified Questions, Answers &
Rationales. - 119 Questions and Answers Already Graded A+
Premium Exam Tested And Verified
Subject Area Industrial and Systems Engineering - Simulation and Stochastic Modeling
Description This exam covers the first half of ISYE 6644, focusing on random number
generation, Monte Carlo methods, input modeling, variance reduction techniques,
and output analysis for discrete-event simulation. Questions require deep
theoretical understanding and application of advanced simulation concepts.
Expected Grade A+
Total Questions 119
Duration 3 hours
Learning Outcomes 1. Apply inverse transform and acceptance-rejection methods for generating
random variates.
2. Analyze and select appropriate input distributions using goodness-of-fit tests.
3. Implement variance reduction techniques such as antithetic variates and control
variates.
4. Design and interpret output analysis for terminating and steady-state
simulations.
5. Evaluate and compare random number generators based on statistical
properties.
Accreditation ABET-accredited program; exam meets standards for graduate-level simulation
courses at R1 universities.
Page 1
,1. A simulation analyst generates 1000 independent observations from an
exponential distribution with mean 2 using the inverse transform method. The
sample mean is 2.05. To test the null hypothesis that the data are exponential with
mean 2, which of the following is the most appropriate test statistic and its
approximate distribution under the null?
A. Kolmogorov-Smirnov test statistic; distribution depends on estimated parameters.
B. Anderson-Darling test statistic; asymptotic distribution is known but critical values are
tabulated.
C. Chi-square goodness-of-fit test with 10 bins; approximately chi-square with 9 degrees of
freedom.
D. t-test for the mean; approximately t-distribution with 999 degrees of freedom.
Answer: B. Anderson-Darling test statistic; asymptotic distribution is known but
critical values are tabulated.
The Anderson-Darling test is more powerful than Kolmogorov-Smirnov for detecting
tail deviations and is appropriate for continuous distributions. The chi-square test
would lose power due to binning. The t-test only checks the mean, not the distributional
form. Option B correctly identifies a standard approach for exponential goodness-of-fit.
2. Consider a linear congruential generator (LCG) with parameters: modulus m =
2^31 - 1, multiplier a = 7^5, increment c = 0, and seed X0 = 1. Which of the following
statements about this generator is correct?
A. It has full period m-1 and passes all spectral tests for dimensions up to 6.
B. It has period at most m-1 but may have poor lattice structure in high dimensions.
C. It is a multiplicative LCG with period exactly m, and it is recommended for all simulation
applications.
D. It fails the chi-square test for uniformity and should not be used.
Answer: B. It has period at most m-1 but may have poor lattice structure in high
dimensions.
This is the classic Lewis-Good-Miller generator (a=16807, m=2^31-1). It has full period
m-1 (since c=0, period divides m-1) and passes many tests, but its lattice structure in
high dimensions is known to be poor. Option A overstates spectral test performance; C
is wrong because period is m-1, not m; D is false as it passes uniformity tests.
Page 2
,3. A simulation of a queueing system uses antithetic variates to estimate the expected
waiting time. Two independent runs produce estimates W1 and W2, each based on
100 replications. The analyst computes the antithetic estimator as (W1 + W2)/2.
Which condition must hold for this estimator to have lower variance than the
average of two independent runs?
A. The covariance between W1 and W2 is negative.
B. The covariance between W1 and W2 is positive.
C. The variance of W1 equals the variance of W2.
D. The runs are based on antithetic random number streams.
Answer: A. The covariance between W1 and W2 is negative.
Variance of the average is (Var(W1)+Var(W2)+2Cov(W1,W2))/4. For antithetic
variates, we induce negative correlation, so Cov<0 reduces variance. Positive
correlation would increase variance. Equal variances are not required but help. Option
D is true but not the condition for variance reduction; the key is negative covariance.
4. In a simulation of a manufacturing system, the analyst uses the batch means
method to estimate the steady-state mean cycle time. The batch size is b and the
number of batches is k. Which of the following is a critical assumption for the batch
means method to produce valid confidence intervals?
A. The batch means are independent and identically distributed normal random variables.
B. The batch size b is large enough that the batch means are approximately uncorrelated.
C. The number of batches k must be at least 30 to apply the central limit theorem.
D. The underlying process must be a stationary AR(1) process.
Answer: B. The batch size b is large enough that the batch means are
approximately uncorrelated.
The batch means method relies on the batch means being approximately uncorrelated
for large b, so that the sample variance of batch means estimates the variance of the
mean. Normality is not required for confidence intervals based on the t-distribution if b
is large enough. Option A is too strong; C is not a fixed rule; D is not necessary.
Page 3
, 5. A simulation analyst is modeling the arrival process to a call center. Historical
data show that the number of calls per hour follows a Poisson distribution with a
rate that varies by time of day. Which of the following approaches is most
appropriate for generating arrival times in the simulation?
A. Use a homogeneous Poisson process with the average daily rate.
B. Use a nonhomogeneous Poisson process with a piecewise constant rate function.
C. Generate interarrival times from an exponential distribution with the overall mean.
D. Use a renewal process with gamma-distributed interarrival times.
Answer: B. Use a nonhomogeneous Poisson process with a piecewise constant rate
function.
A nonhomogeneous Poisson process captures time-varying arrival rates, which is
essential for call centers. Option A ignores rate variation; C is equivalent to A; D
assumes a renewal process but does not handle time-dependent rates. Thinning or
inversion methods can generate nonhomogeneous Poisson processes.
6. Consider the following algorithm for generating a random variate from a
distribution with CDF F(x) = 1 - exp(-x^2) for x >= 0. Which method is most
efficient?
A. Inverse transform: set X = sqrt(-ln(1-U)) where U ~ Uniform(0,1).
B. Acceptance-rejection using an exponential envelope with rate 1.
C. Box-Muller transform to generate Rayleigh variates.
D. Composition method by mixing two exponential distributions.
Answer: A. Inverse transform: set X = sqrt(-ln(1-U)) where U ~ Uniform(0,1).
The inverse CDF is easily derived: solving F(x)=u gives x = sqrt(-ln(1-u)). This is direct
and efficient. Acceptance-rejection would be less efficient. Box-Muller generates normal
variates, not this distribution. Composition is not needed when a simple inverse exists.
Page 4
Complete Study Guide with Verified Questions, Answers &
Rationales. - 119 Questions and Answers Already Graded A+
Premium Exam Tested And Verified
Subject Area Industrial and Systems Engineering - Simulation and Stochastic Modeling
Description This exam covers the first half of ISYE 6644, focusing on random number
generation, Monte Carlo methods, input modeling, variance reduction techniques,
and output analysis for discrete-event simulation. Questions require deep
theoretical understanding and application of advanced simulation concepts.
Expected Grade A+
Total Questions 119
Duration 3 hours
Learning Outcomes 1. Apply inverse transform and acceptance-rejection methods for generating
random variates.
2. Analyze and select appropriate input distributions using goodness-of-fit tests.
3. Implement variance reduction techniques such as antithetic variates and control
variates.
4. Design and interpret output analysis for terminating and steady-state
simulations.
5. Evaluate and compare random number generators based on statistical
properties.
Accreditation ABET-accredited program; exam meets standards for graduate-level simulation
courses at R1 universities.
Page 1
,1. A simulation analyst generates 1000 independent observations from an
exponential distribution with mean 2 using the inverse transform method. The
sample mean is 2.05. To test the null hypothesis that the data are exponential with
mean 2, which of the following is the most appropriate test statistic and its
approximate distribution under the null?
A. Kolmogorov-Smirnov test statistic; distribution depends on estimated parameters.
B. Anderson-Darling test statistic; asymptotic distribution is known but critical values are
tabulated.
C. Chi-square goodness-of-fit test with 10 bins; approximately chi-square with 9 degrees of
freedom.
D. t-test for the mean; approximately t-distribution with 999 degrees of freedom.
Answer: B. Anderson-Darling test statistic; asymptotic distribution is known but
critical values are tabulated.
The Anderson-Darling test is more powerful than Kolmogorov-Smirnov for detecting
tail deviations and is appropriate for continuous distributions. The chi-square test
would lose power due to binning. The t-test only checks the mean, not the distributional
form. Option B correctly identifies a standard approach for exponential goodness-of-fit.
2. Consider a linear congruential generator (LCG) with parameters: modulus m =
2^31 - 1, multiplier a = 7^5, increment c = 0, and seed X0 = 1. Which of the following
statements about this generator is correct?
A. It has full period m-1 and passes all spectral tests for dimensions up to 6.
B. It has period at most m-1 but may have poor lattice structure in high dimensions.
C. It is a multiplicative LCG with period exactly m, and it is recommended for all simulation
applications.
D. It fails the chi-square test for uniformity and should not be used.
Answer: B. It has period at most m-1 but may have poor lattice structure in high
dimensions.
This is the classic Lewis-Good-Miller generator (a=16807, m=2^31-1). It has full period
m-1 (since c=0, period divides m-1) and passes many tests, but its lattice structure in
high dimensions is known to be poor. Option A overstates spectral test performance; C
is wrong because period is m-1, not m; D is false as it passes uniformity tests.
Page 2
,3. A simulation of a queueing system uses antithetic variates to estimate the expected
waiting time. Two independent runs produce estimates W1 and W2, each based on
100 replications. The analyst computes the antithetic estimator as (W1 + W2)/2.
Which condition must hold for this estimator to have lower variance than the
average of two independent runs?
A. The covariance between W1 and W2 is negative.
B. The covariance between W1 and W2 is positive.
C. The variance of W1 equals the variance of W2.
D. The runs are based on antithetic random number streams.
Answer: A. The covariance between W1 and W2 is negative.
Variance of the average is (Var(W1)+Var(W2)+2Cov(W1,W2))/4. For antithetic
variates, we induce negative correlation, so Cov<0 reduces variance. Positive
correlation would increase variance. Equal variances are not required but help. Option
D is true but not the condition for variance reduction; the key is negative covariance.
4. In a simulation of a manufacturing system, the analyst uses the batch means
method to estimate the steady-state mean cycle time. The batch size is b and the
number of batches is k. Which of the following is a critical assumption for the batch
means method to produce valid confidence intervals?
A. The batch means are independent and identically distributed normal random variables.
B. The batch size b is large enough that the batch means are approximately uncorrelated.
C. The number of batches k must be at least 30 to apply the central limit theorem.
D. The underlying process must be a stationary AR(1) process.
Answer: B. The batch size b is large enough that the batch means are
approximately uncorrelated.
The batch means method relies on the batch means being approximately uncorrelated
for large b, so that the sample variance of batch means estimates the variance of the
mean. Normality is not required for confidence intervals based on the t-distribution if b
is large enough. Option A is too strong; C is not a fixed rule; D is not necessary.
Page 3
, 5. A simulation analyst is modeling the arrival process to a call center. Historical
data show that the number of calls per hour follows a Poisson distribution with a
rate that varies by time of day. Which of the following approaches is most
appropriate for generating arrival times in the simulation?
A. Use a homogeneous Poisson process with the average daily rate.
B. Use a nonhomogeneous Poisson process with a piecewise constant rate function.
C. Generate interarrival times from an exponential distribution with the overall mean.
D. Use a renewal process with gamma-distributed interarrival times.
Answer: B. Use a nonhomogeneous Poisson process with a piecewise constant rate
function.
A nonhomogeneous Poisson process captures time-varying arrival rates, which is
essential for call centers. Option A ignores rate variation; C is equivalent to A; D
assumes a renewal process but does not handle time-dependent rates. Thinning or
inversion methods can generate nonhomogeneous Poisson processes.
6. Consider the following algorithm for generating a random variate from a
distribution with CDF F(x) = 1 - exp(-x^2) for x >= 0. Which method is most
efficient?
A. Inverse transform: set X = sqrt(-ln(1-U)) where U ~ Uniform(0,1).
B. Acceptance-rejection using an exponential envelope with rate 1.
C. Box-Muller transform to generate Rayleigh variates.
D. Composition method by mixing two exponential distributions.
Answer: A. Inverse transform: set X = sqrt(-ln(1-U)) where U ~ Uniform(0,1).
The inverse CDF is easily derived: solving F(x)=u gives x = sqrt(-ln(1-u)). This is direct
and efficient. Acceptance-rejection would be less efficient. Box-Muller generates normal
variates, not this distribution. Composition is not needed when a simple inverse exists.
Page 4
