ISYE 6644 Simulation Ultimate
Final Exam Bank (Questions1-400)
| Georgia Tech | 400 Q&As with
Verified Answers, Step-by-Step
Mathematical Solutions &
Highlightable Rationales
This master study bank features 50 exam-style
multiple-choice questions for the Georgia Tech ISYE
6644 Simulation curriculum [Questions1-400].
Meticulously formatted with correct options and
comprehensive rationales in bold-italic text, this file
allows for text scanning and digital copy-pasting.
Build total clarity on core concepts including the
Inverse Transform Method, LCG tracking, queueing
theory formulas, variance reduction techniques,
Arena advanced modules, and output analysis
mechanisms to optimize your academic preparation
and Stuvia document earnings.
,1. Suppose X is a continuous random variable with a probability density function
(pdf) given by f(x) = 3x² for 0 ≤ x ≤ 1, and f(x) = 0 otherwise. Using the Inverse
Transform Method, find the inverse CDF formula X = F⁻¹(U) to generate samples of
X from a Uniform(0, 1) random variable U.
A) X = U³
B) \(X = U^{1/3}\)
C) X = 1 - U³
D) \(X = 3U^{1/3}\)
Correct Answer: B
Rationale: To implement the Inverse Transform Method, we first find the Cumulative
Distribution Function (CDF), F(x), by integrating the pdf: \(F(x) = \int_{0}^{x} 3t^2 dt =
x^3\) for 0 ≤ x ≤ 1. Next, we set F(x) = U, which yields x³ = U. Solving for x gives the
inverse function: \(X = U^{1/3}\). This allows direct mapping of uniform random numbers
to the target distribution.
2. A Linear Congruential Generator (LCG) is defined by the recurrence relation
\(X_{i} = (5X_{i-1} + 3) \mod 8\). If the seed value is X₀ = 2, calculate the pseudo-
random values generated for the first two iterations, U₁ and U₂.
A) U₁ = 0.500, U₂ = 0.250
B) U₁ = 0.625, U₂ = 0.500
C) U₁ = 0.625, U₂ = 0.750
D) U₁ = 0.125, U₂ = 0.375
Correct Answer: B
Rationale: First iteration: \(X_1 = (5 \times 2 + 3) \mod 8 = 13 \mod 8 = 5\). The
normalized uniform value is U₁ = X₁ / m = = 0.625. Second iteration: \(X_2 = (5
\times 5 + 3) \mod 8 = 28 \mod 8 = 4\). The normalized uniform value is U₂ = X₂ / m = 4
/ 8 = 0.500.
3. In a single-server queueing simulation (M/M/1), customers arrive at a rate of λ =
4 per hour and are served at a rate of μ = 5 per hour. What is the steady-state
expected number of customers L in the system?
A) 0.8
B) 1.25
C) 4.0
D) 5.0
Correct Answer: C
Rationale: For an M/M/1 queue, the traffic intensity is ρ = λ / μ = 4/5 = 0.8. The steady-
state expected number of customers in the system L is calculated using the formula L =
ρ / (1 - ρ). Substituting the values yields L = 0.8 / (1 - 0.8) = 0..2 = 4.0.
4. When conducting output analysis for a steady-state simulation, which method
is primarily used to mitigate the initialization bias caused by the "warm-up"
,period?
A) Antithetic Variates
B) Welch’s Graphical Procedure
C) Common Random Numbers (CRN)
D) Chi-Square Goodness-of-Fit Test
Correct Answer: B
Rationale: Initialization bias occurs when a simulation starts from an unrealistic state
(like completely empty and idle), skewing steady-state estimates. Welch’s Graphical
Procedure is a visual approach used to determine the length of the warm-up period by
plotting moving averages of the output data over multiple replications, allowing
researchers to safely truncate the biased initial observations.
5. Suppose you are using the Acceptance-Rejection method to generate samples
from a target distribution f(x) using a majorizing function e(x) = c ⋅ g(x), where g(x)
is a valid proposal density. If the constant c = 2.5, what is the expected number of
iterations required to successfully accept a single random variable sample?
A) 1.0
B) 2.5
C) 0.4
D) 6.25
Correct Answer: B
Rationale: In the Acceptance-Rejection algorithm, the probability of accepting a
candidate sample on any given trial is equal to 1/c. Because each trial is an
independent Bernoulli trial, the total number of iterations required to achieve the first
success follows a Geometric distribution with an expected value of 1 / (1/c) = c.
Therefore, the expected number of iterations is exactly c = 2.5.
6. Which of the following conditions represents the Hull-Dobell Theorem
requirements for a Linear Congruential Generator (LCG) of the form \(X_i = (aX_{i-
1} + c) \mod m\) to achieve a full period of length m?
A) c and m are relatively prime; a-1 is divisible by all prime factors of m; a-1 is divisible
by 4 if 4 divides m.
B) m is a prime number; c = 0; a is a primitive root of m.
C) a is a multiple of c; m is a power of 2; c is odd.
D) c = 0; a-1 is divisible by all prime factors of m; m is relatively prime to X₀.
Correct Answer: A
Rationale: The Hull-Dobell Theorem provides the exact necessary and sufficient
conditions for a mixed LCG (where c > 0) to have a full period of length m. The three
strict conditions are: 1) c and m must be relatively prime (gcd(c,m)=1); 2) a-1 must be
divisible by every prime factor of m; 3) a-1 must be a multiple of 4 if m is a multiple of 4.
7. Use a 1-step Monte Carlo integration calculation to estimate the integral \(I =
\int_{0}^{2} 4x^3 dx\) using a single uniform random number value U = 0.5.
A) 2.0
, B) 4.0
C) 8.0
D) 16.0
Correct Answer: C
Rationale: To estimate an integral over [a, b], we rewrite it as \(I = (b - a)
\mathbb{E}[f(X)]\), where X ~ Uniform(a, b). For U = 0.5, we map it to [0, 2] via X = a +
(b - a)U = 0 + (2 - 0)(0.5) = 1.0. Evaluate the function at X = 1: f(1) = 4(1)³ = 4. The 1-
step Monte Carlo estimate is (b - a) × f(X) = (2 - 0) × 4 = 8.0.
8. Which variance reduction technique reduces output variance by using pairs of
negatively correlated random numbers within the same simulation model?
A) Common Random Numbers (CRN)
B) Control Variates
C) Antithetic Variates
D) Importance Sampling
Correct Answer: C
Rationale: Antithetic Variates relies on driving a simulation using negatively correlated
inputs. If a run uses uniform random variables \(U_1, U_2, \dots\), a paired run is
executed using \(1-U_1, 1-U_2, \dots\). When these two outputs are averaged, their
negative correlation causes their individual variations about the true mean to cancel out,
substantially reducing the overall variance of the estimator.
9. In Arena simulation software, which template module is fundamentally used to
define the process parameters, process times, and resource allocations for an
entity passing through a system?
A) Create Module
B) Decide Module
C) Process Module
D) Dispose Module
Correct Answer: C
Rationale: The Process Module is the core functional logic element in Arena where
entities encounter delay, seize resources, and release resources. It allows users to
define the duration distributions (e.g., Triangular, Normal) and assign the specific logic
types such as "Seize Delay Release," "Delay Only," or "Seize Delay."
10. Suppose we want to conduct a Chi-Square Goodness-of-Fit test to see if our
data fits a Poisson distribution. The data is grouped into k = 6 bins, and we must
estimate the parameter λ from the data sample. What are the correct degrees of
freedom for our test statistic?
A) 6
B) 5
C) 4
D) 3
Correct Answer: C
Final Exam Bank (Questions1-400)
| Georgia Tech | 400 Q&As with
Verified Answers, Step-by-Step
Mathematical Solutions &
Highlightable Rationales
This master study bank features 50 exam-style
multiple-choice questions for the Georgia Tech ISYE
6644 Simulation curriculum [Questions1-400].
Meticulously formatted with correct options and
comprehensive rationales in bold-italic text, this file
allows for text scanning and digital copy-pasting.
Build total clarity on core concepts including the
Inverse Transform Method, LCG tracking, queueing
theory formulas, variance reduction techniques,
Arena advanced modules, and output analysis
mechanisms to optimize your academic preparation
and Stuvia document earnings.
,1. Suppose X is a continuous random variable with a probability density function
(pdf) given by f(x) = 3x² for 0 ≤ x ≤ 1, and f(x) = 0 otherwise. Using the Inverse
Transform Method, find the inverse CDF formula X = F⁻¹(U) to generate samples of
X from a Uniform(0, 1) random variable U.
A) X = U³
B) \(X = U^{1/3}\)
C) X = 1 - U³
D) \(X = 3U^{1/3}\)
Correct Answer: B
Rationale: To implement the Inverse Transform Method, we first find the Cumulative
Distribution Function (CDF), F(x), by integrating the pdf: \(F(x) = \int_{0}^{x} 3t^2 dt =
x^3\) for 0 ≤ x ≤ 1. Next, we set F(x) = U, which yields x³ = U. Solving for x gives the
inverse function: \(X = U^{1/3}\). This allows direct mapping of uniform random numbers
to the target distribution.
2. A Linear Congruential Generator (LCG) is defined by the recurrence relation
\(X_{i} = (5X_{i-1} + 3) \mod 8\). If the seed value is X₀ = 2, calculate the pseudo-
random values generated for the first two iterations, U₁ and U₂.
A) U₁ = 0.500, U₂ = 0.250
B) U₁ = 0.625, U₂ = 0.500
C) U₁ = 0.625, U₂ = 0.750
D) U₁ = 0.125, U₂ = 0.375
Correct Answer: B
Rationale: First iteration: \(X_1 = (5 \times 2 + 3) \mod 8 = 13 \mod 8 = 5\). The
normalized uniform value is U₁ = X₁ / m = = 0.625. Second iteration: \(X_2 = (5
\times 5 + 3) \mod 8 = 28 \mod 8 = 4\). The normalized uniform value is U₂ = X₂ / m = 4
/ 8 = 0.500.
3. In a single-server queueing simulation (M/M/1), customers arrive at a rate of λ =
4 per hour and are served at a rate of μ = 5 per hour. What is the steady-state
expected number of customers L in the system?
A) 0.8
B) 1.25
C) 4.0
D) 5.0
Correct Answer: C
Rationale: For an M/M/1 queue, the traffic intensity is ρ = λ / μ = 4/5 = 0.8. The steady-
state expected number of customers in the system L is calculated using the formula L =
ρ / (1 - ρ). Substituting the values yields L = 0.8 / (1 - 0.8) = 0..2 = 4.0.
4. When conducting output analysis for a steady-state simulation, which method
is primarily used to mitigate the initialization bias caused by the "warm-up"
,period?
A) Antithetic Variates
B) Welch’s Graphical Procedure
C) Common Random Numbers (CRN)
D) Chi-Square Goodness-of-Fit Test
Correct Answer: B
Rationale: Initialization bias occurs when a simulation starts from an unrealistic state
(like completely empty and idle), skewing steady-state estimates. Welch’s Graphical
Procedure is a visual approach used to determine the length of the warm-up period by
plotting moving averages of the output data over multiple replications, allowing
researchers to safely truncate the biased initial observations.
5. Suppose you are using the Acceptance-Rejection method to generate samples
from a target distribution f(x) using a majorizing function e(x) = c ⋅ g(x), where g(x)
is a valid proposal density. If the constant c = 2.5, what is the expected number of
iterations required to successfully accept a single random variable sample?
A) 1.0
B) 2.5
C) 0.4
D) 6.25
Correct Answer: B
Rationale: In the Acceptance-Rejection algorithm, the probability of accepting a
candidate sample on any given trial is equal to 1/c. Because each trial is an
independent Bernoulli trial, the total number of iterations required to achieve the first
success follows a Geometric distribution with an expected value of 1 / (1/c) = c.
Therefore, the expected number of iterations is exactly c = 2.5.
6. Which of the following conditions represents the Hull-Dobell Theorem
requirements for a Linear Congruential Generator (LCG) of the form \(X_i = (aX_{i-
1} + c) \mod m\) to achieve a full period of length m?
A) c and m are relatively prime; a-1 is divisible by all prime factors of m; a-1 is divisible
by 4 if 4 divides m.
B) m is a prime number; c = 0; a is a primitive root of m.
C) a is a multiple of c; m is a power of 2; c is odd.
D) c = 0; a-1 is divisible by all prime factors of m; m is relatively prime to X₀.
Correct Answer: A
Rationale: The Hull-Dobell Theorem provides the exact necessary and sufficient
conditions for a mixed LCG (where c > 0) to have a full period of length m. The three
strict conditions are: 1) c and m must be relatively prime (gcd(c,m)=1); 2) a-1 must be
divisible by every prime factor of m; 3) a-1 must be a multiple of 4 if m is a multiple of 4.
7. Use a 1-step Monte Carlo integration calculation to estimate the integral \(I =
\int_{0}^{2} 4x^3 dx\) using a single uniform random number value U = 0.5.
A) 2.0
, B) 4.0
C) 8.0
D) 16.0
Correct Answer: C
Rationale: To estimate an integral over [a, b], we rewrite it as \(I = (b - a)
\mathbb{E}[f(X)]\), where X ~ Uniform(a, b). For U = 0.5, we map it to [0, 2] via X = a +
(b - a)U = 0 + (2 - 0)(0.5) = 1.0. Evaluate the function at X = 1: f(1) = 4(1)³ = 4. The 1-
step Monte Carlo estimate is (b - a) × f(X) = (2 - 0) × 4 = 8.0.
8. Which variance reduction technique reduces output variance by using pairs of
negatively correlated random numbers within the same simulation model?
A) Common Random Numbers (CRN)
B) Control Variates
C) Antithetic Variates
D) Importance Sampling
Correct Answer: C
Rationale: Antithetic Variates relies on driving a simulation using negatively correlated
inputs. If a run uses uniform random variables \(U_1, U_2, \dots\), a paired run is
executed using \(1-U_1, 1-U_2, \dots\). When these two outputs are averaged, their
negative correlation causes their individual variations about the true mean to cancel out,
substantially reducing the overall variance of the estimator.
9. In Arena simulation software, which template module is fundamentally used to
define the process parameters, process times, and resource allocations for an
entity passing through a system?
A) Create Module
B) Decide Module
C) Process Module
D) Dispose Module
Correct Answer: C
Rationale: The Process Module is the core functional logic element in Arena where
entities encounter delay, seize resources, and release resources. It allows users to
define the duration distributions (e.g., Triangular, Normal) and assign the specific logic
types such as "Seize Delay Release," "Delay Only," or "Seize Delay."
10. Suppose we want to conduct a Chi-Square Goodness-of-Fit test to see if our
data fits a Poisson distribution. The data is grouped into k = 6 bins, and we must
estimate the parameter λ from the data sample. What are the correct degrees of
freedom for our test statistic?
A) 6
B) 5
C) 4
D) 3
Correct Answer: C
