GRADUATE ENGINEERING
ISYE 6644 - Summer 2026/2027 - Final Exam Questions and
Answers
Complete Blueprint Coverage A+
A+ 5 100%
QUESTIONS VERIFIED CORE DOMAINS COVERED RATIONALES INCLUDED
CATEGORIES
Random Number Generation and Monte Carlo Methods
Discrete-Event Simulation Modeling
Input Modeling and Distribution Fitting
Output Analysis and Variance Reduction
Advanced Simulation Topics
STUVIAACTUALEXAM
,ISYE 6644 - Summer 2026/2027 - Final Exam Questions and Answers Page 2
RANDOM NUMBER GENERATION AND MONTE CARLO METHODS
Q1
A simulation study requires a stream of uniform(0,1) random numbers. The analyst implements a linear congruential
generator with parameters a = 16,807, m = 2^31 - 1, and c = 0. After generating 50,000 numbers the analyst
observes that the period appears shorter than expected for full-period behavior. The most likely cause of the reduced
period is:
A. The multiplier and modulus are relatively prime but the initial seed was chosen poorly, violating the Hull-Dobell theorem
conditions for full period
B. The modulus 2^31-1 is composite, guaranteeing a short period regardless of the multiplier
C. Floating-point precision errors accumulate and force the sequence to cycle early
D. The constant c = 0 makes the generator multiplicative; full period is impossible and the maximum achievable period is
m-1 only under specific multiplier conditions
Correct Answer: D
When c = 0 the generator is multiplicative. Full period m is impossible; the maximum period is at most m-1 and is attained only for
carefully chosen multipliers. The given modulus is prime, so the short observed period is attributable to the multiplicative form
rather than a composite modulus or seed alone.
Q2
An analyst needs to generate a random variate from the continuous distribution with density f(x) = 2x on [0,1]. Using
the inverse-transform method and a single U ~ Uniform(0,1), the correct transformation is:
A. X = U^2
B. X = -ln(U)
C. X = sqrt(U)
D. X = 1 - U
Correct Answer: C
The CDF is F(x) = x^2 on [0,1]. Inverting U = X^2 yields X = sqrt(U). The other transformations correspond to different distributions
(power, exponential, or uniform).
Q3
A Monte Carlo experiment estimates the integral of a bounded function g over the unit interval by the sample-mean
estimator. After n independent replications the analyst constructs a 95 percent confidence interval. Doubling the
number of replications while keeping everything else fixed is expected to:
A. Halve the half-width of the confidence interval
B. Reduce the half-width by a factor of approximately sqrt(2)
C. Leave the half-width unchanged because the variance of g is fixed
D. Double the half-width because more data increase uncertainty
Correct Answer: B
The standard error of the sample mean decreases as 1/sqrt(n). Doubling n multiplies the standard error (and therefore the
confidence-interval half-width) by 1/sqrt(2).
STUVIAACTUALEXAM
, ISYE 6644 - Summer 2026/2027 - Final Exam Questions and Answers Page 3
Q4
Two random-number streams are needed for a simulation that compares two alternative system designs. The analyst
wants the same random numbers to drive corresponding parts of both systems so that differences are attributable to
design rather than random noise. The correct technique is:
A. Common random numbers (synchronization of random-number streams across alternatives)
B. Antithetic variates applied independently within each alternative
C. Independent seeding of every random-number generator call
D. Latin-hypercube sampling of the input parameters only
Correct Answer: A
Common random numbers induce positive correlation between the outputs of the two systems, thereby reducing the variance of the
difference estimator. Antithetic variates and independent seeding do not create the desired cross-design synchronization.
Q5
An acceptance-rejection algorithm is used to generate random variates from a target density f that is difficult to invert.
The majorizing function is cg(x) where g is an easy density. The expected number of trials until the first acceptance
equals:
A. 1/c
B. c
C. c^2
D. The integral of f
Correct Answer: B
The acceptance probability on each independent trial is 1/c. Consequently the number of trials until the first acceptance follows a
geometric distribution with success probability 1/c, whose expectation is c.
Q6
A simulation requires non-uniform random numbers from a discrete distribution with probability mass function p(i) for i
= 1,...,k. The analyst implements the inverse-transform method by computing the cumulative probabilities and
comparing a Uniform(0,1) draw against the cumulative table. The computational complexity per variate in the worst
case is:
A. O(1) regardless of the search method
B. O(k) if a sequential search is used, or O(log k) if binary search is used on the cumulative table
C. O(k^2) because each cumulative probability must be recomputed
D. O(n) where n is the sample size already generated
Correct Answer: B
Building the cumulative distribution is done once. Each subsequent generation requires a search of the cumulative table; sequential
search costs O(k) and binary search costs O(log k).
STUVIAACTUALEXAM
ISYE 6644 - Summer 2026/2027 - Final Exam Questions and
Answers
Complete Blueprint Coverage A+
A+ 5 100%
QUESTIONS VERIFIED CORE DOMAINS COVERED RATIONALES INCLUDED
CATEGORIES
Random Number Generation and Monte Carlo Methods
Discrete-Event Simulation Modeling
Input Modeling and Distribution Fitting
Output Analysis and Variance Reduction
Advanced Simulation Topics
STUVIAACTUALEXAM
,ISYE 6644 - Summer 2026/2027 - Final Exam Questions and Answers Page 2
RANDOM NUMBER GENERATION AND MONTE CARLO METHODS
Q1
A simulation study requires a stream of uniform(0,1) random numbers. The analyst implements a linear congruential
generator with parameters a = 16,807, m = 2^31 - 1, and c = 0. After generating 50,000 numbers the analyst
observes that the period appears shorter than expected for full-period behavior. The most likely cause of the reduced
period is:
A. The multiplier and modulus are relatively prime but the initial seed was chosen poorly, violating the Hull-Dobell theorem
conditions for full period
B. The modulus 2^31-1 is composite, guaranteeing a short period regardless of the multiplier
C. Floating-point precision errors accumulate and force the sequence to cycle early
D. The constant c = 0 makes the generator multiplicative; full period is impossible and the maximum achievable period is
m-1 only under specific multiplier conditions
Correct Answer: D
When c = 0 the generator is multiplicative. Full period m is impossible; the maximum period is at most m-1 and is attained only for
carefully chosen multipliers. The given modulus is prime, so the short observed period is attributable to the multiplicative form
rather than a composite modulus or seed alone.
Q2
An analyst needs to generate a random variate from the continuous distribution with density f(x) = 2x on [0,1]. Using
the inverse-transform method and a single U ~ Uniform(0,1), the correct transformation is:
A. X = U^2
B. X = -ln(U)
C. X = sqrt(U)
D. X = 1 - U
Correct Answer: C
The CDF is F(x) = x^2 on [0,1]. Inverting U = X^2 yields X = sqrt(U). The other transformations correspond to different distributions
(power, exponential, or uniform).
Q3
A Monte Carlo experiment estimates the integral of a bounded function g over the unit interval by the sample-mean
estimator. After n independent replications the analyst constructs a 95 percent confidence interval. Doubling the
number of replications while keeping everything else fixed is expected to:
A. Halve the half-width of the confidence interval
B. Reduce the half-width by a factor of approximately sqrt(2)
C. Leave the half-width unchanged because the variance of g is fixed
D. Double the half-width because more data increase uncertainty
Correct Answer: B
The standard error of the sample mean decreases as 1/sqrt(n). Doubling n multiplies the standard error (and therefore the
confidence-interval half-width) by 1/sqrt(2).
STUVIAACTUALEXAM
, ISYE 6644 - Summer 2026/2027 - Final Exam Questions and Answers Page 3
Q4
Two random-number streams are needed for a simulation that compares two alternative system designs. The analyst
wants the same random numbers to drive corresponding parts of both systems so that differences are attributable to
design rather than random noise. The correct technique is:
A. Common random numbers (synchronization of random-number streams across alternatives)
B. Antithetic variates applied independently within each alternative
C. Independent seeding of every random-number generator call
D. Latin-hypercube sampling of the input parameters only
Correct Answer: A
Common random numbers induce positive correlation between the outputs of the two systems, thereby reducing the variance of the
difference estimator. Antithetic variates and independent seeding do not create the desired cross-design synchronization.
Q5
An acceptance-rejection algorithm is used to generate random variates from a target density f that is difficult to invert.
The majorizing function is cg(x) where g is an easy density. The expected number of trials until the first acceptance
equals:
A. 1/c
B. c
C. c^2
D. The integral of f
Correct Answer: B
The acceptance probability on each independent trial is 1/c. Consequently the number of trials until the first acceptance follows a
geometric distribution with success probability 1/c, whose expectation is c.
Q6
A simulation requires non-uniform random numbers from a discrete distribution with probability mass function p(i) for i
= 1,...,k. The analyst implements the inverse-transform method by computing the cumulative probabilities and
comparing a Uniform(0,1) draw against the cumulative table. The computational complexity per variate in the worst
case is:
A. O(1) regardless of the search method
B. O(k) if a sequential search is used, or O(log k) if binary search is used on the cumulative table
C. O(k^2) because each cumulative probability must be recomputed
D. O(n) where n is the sample size already generated
Correct Answer: B
Building the cumulative distribution is done once. Each subsequent generation requires a search of the cumulative table; sequential
search costs O(k) and binary search costs O(log k).
STUVIAACTUALEXAM
