ISYE 6644 - Summer 2024 - Final Exam
Questions and Answers with complete
solution
SECTION 1: SIMULATION FUNDAMENTALS & PROBABILITY REVIEW
Question 1
Which of the following best describes the primary difference between verification and
validation in a simulation study?
A. Verification checks if the model output matches real-world data; validation checks if the code
runs without errors
B. Verification asks "Are we building the model right?"; validation asks "Are we building
the right model?"
C. Verification is performed by stakeholders; validation is performed by developers
D. Verification is optional for complex models; validation is always required
Rationale: Verification is the process of determining whether a simulation program correctly
implements the conceptual model—essentially debugging and confirming logical correctness.
Validation determines whether the model accurately represents the real system being studied.
Verification precedes validation, and both are essential for credible simulation results .
Question 2 [SATA]
Which of the following are characteristics of a discrete-event simulation (DES)? [Select all that
apply]
A. System state changes occur at discrete points in time
B. State variables change continuously over time
C. The simulation clock advances from event to event
D. Entities flow through queues and resources
E. Time between events may be stochastic
, Rationale: DES models systems where state changes occur at discrete, countable time
points driven by events (arrivals, service completions, departures). The clock jumps between
event times rather than advancing continuously. Entities (customers, parts) flow through
processes involving queues, resources, and delays. The timing of events is typically stochastic in
real-world applications .
Question 3
A bank drive-through teller system is best modeled using which type of simulation?
A. Continuous simulation
B. Discrete-event simulation
C. Static Monte Carlo simulation
D. Deterministic differential equation model
Rationale: A bank drive-through is a classic discrete-event system: state changes (customer
arrival, service start, service completion, departure) occur at distinct points in time. Between
events, the system state remains constant. Continuous simulation is inappropriate because the
number of customers in queue changes discretely, not continuously .
Question 4
1
If 𝑋and 𝑌have joint p.d.f. 𝑓(𝑥, 𝑦) = 2for 0 ≤ 𝑥 ≤ 1,0 ≤ 𝑦 ≤ 2, what is 𝑃(𝑌 > 𝑋)?
A. 1/4
B. 1/2
C. 3/4
D. 2/3
Rationale: The joint p.d.f. is uniform over the rectangle 0 ≤ 𝑥 ≤ 1,0 ≤ 𝑦 ≤ 2. The region
1 1
where 𝑌 > 𝑋excludes only the triangle where 𝑌 ≤ 𝑋. The area of the triangle is 2 ⋅ 1 ⋅ 1 = 2.
2−1/2 3/2 3
The rectangle area is 2. Probability = = = 4.
2 2
Question 5
1
Are 𝑋and 𝑌independent if 𝑓(𝑥, 𝑦) = 2for 0 ≤ 𝑥 ≤ 1,0 ≤ 𝑦 ≤ 2?
,A. Yes
B. No
Rationale: The joint p.d.f. factors as 𝑓(𝑥, 𝑦) = 𝑓𝑋 (𝑥) ⋅ 𝑓𝑌 (𝑦)where 𝑓𝑋 (𝑥) = 1for 0 ≤ 𝑥 ≤
1
1and 𝑓𝑌 (𝑦) = 2for 0 ≤ 𝑦 ≤ 2. Since the domain is a rectangle and the joint density factors into
functions of 𝑥and 𝑦separately, 𝑋and 𝑌are independent .
Question 6
If 𝑋and 𝑌are i.i.d. Nor(1, 4) random variables, find Var(3X - 2Y).
A. 0
B. 10
C. 20
D. 52
E. 208
Rationale: Var(3X - 2Y) = 9Var(X) + 4Var(Y) - 12Cov(X,Y). Since X and Y are independent,
Cov(X,Y) = 0. Var(X) = Var(Y) = 4. Thus Var = 9(4) + 4(4) = 36 + 16 = 52 .
Question 7
If 𝑋and 𝑌are i.i.d. Unif(0,1) random variables, find E[2XY].
A. 0
B. 1/2
C. 1/4
D. 1
Rationale: E[2XY] = 2E[X]E[Y] by independence. E[X] = E[Y] = 1/2. Thus E[2XY] = 2(1/2)(1/2) =
1/2 .
Question 8
The covariance of 𝑋and 𝑌is 1/2. What can be concluded?
A. X and Y are positively correlated but could be independent
B. X and Y cannot be independent
, C. X and Y must be independent
D. X and Y have correlation coefficient 1
Rationale: Independence implies zero covariance. Since Cov(X,Y) = 1/2 ≠ 0, X and Y cannot
be independent. Non-zero covariance indicates a linear relationship exists .
Question 9
If 𝑈is Unif(0,1), how can we simulate a Geom(0.6) random variate?
A. [6𝑈] + 1
B. [ln(𝑈)/ln(0.4)]
C. [ln(𝑈)/ln(0.6)]
D. [𝑈/0.6]
Rationale: For geometric distribution with success probability p = 0.6, the inverse transform
method gives 𝑋 = ⌈ln(𝑈)/ln(1 − 𝑝)⌉or equivalently [ln(𝑈)/ln(0.4)](using floor). The
geometric is the number of trials until first success .
Question 10
Suppose 𝑈and 𝑉are PRNs. Let 𝑋 = 𝑈 + 𝑉. As the sample size increases, what p.d.f. does the
histogram of 𝑋approach?
A. Uniform
B. Normal
C. Triangular
D. Exponential
Rationale: The sum of two independent Unif(0,1) random variables follows a symmetric
triangular distribution on [0,2] with mode at 1. This is a classic convolution result .
Question 11 [SATA]
Which of the following are true about acceptance-rejection (A-R) generation? [Select all that
apply]
A. The closer the majorizing function 𝑡(𝑥)is to 𝑓(𝑥), the more efficient A-R is
B. ℎ(𝑦) = 𝑡(𝑦)/ ∫ 𝑡 (𝑥)𝑑𝑥must be a valid p.d.f.
Questions and Answers with complete
solution
SECTION 1: SIMULATION FUNDAMENTALS & PROBABILITY REVIEW
Question 1
Which of the following best describes the primary difference between verification and
validation in a simulation study?
A. Verification checks if the model output matches real-world data; validation checks if the code
runs without errors
B. Verification asks "Are we building the model right?"; validation asks "Are we building
the right model?"
C. Verification is performed by stakeholders; validation is performed by developers
D. Verification is optional for complex models; validation is always required
Rationale: Verification is the process of determining whether a simulation program correctly
implements the conceptual model—essentially debugging and confirming logical correctness.
Validation determines whether the model accurately represents the real system being studied.
Verification precedes validation, and both are essential for credible simulation results .
Question 2 [SATA]
Which of the following are characteristics of a discrete-event simulation (DES)? [Select all that
apply]
A. System state changes occur at discrete points in time
B. State variables change continuously over time
C. The simulation clock advances from event to event
D. Entities flow through queues and resources
E. Time between events may be stochastic
, Rationale: DES models systems where state changes occur at discrete, countable time
points driven by events (arrivals, service completions, departures). The clock jumps between
event times rather than advancing continuously. Entities (customers, parts) flow through
processes involving queues, resources, and delays. The timing of events is typically stochastic in
real-world applications .
Question 3
A bank drive-through teller system is best modeled using which type of simulation?
A. Continuous simulation
B. Discrete-event simulation
C. Static Monte Carlo simulation
D. Deterministic differential equation model
Rationale: A bank drive-through is a classic discrete-event system: state changes (customer
arrival, service start, service completion, departure) occur at distinct points in time. Between
events, the system state remains constant. Continuous simulation is inappropriate because the
number of customers in queue changes discretely, not continuously .
Question 4
1
If 𝑋and 𝑌have joint p.d.f. 𝑓(𝑥, 𝑦) = 2for 0 ≤ 𝑥 ≤ 1,0 ≤ 𝑦 ≤ 2, what is 𝑃(𝑌 > 𝑋)?
A. 1/4
B. 1/2
C. 3/4
D. 2/3
Rationale: The joint p.d.f. is uniform over the rectangle 0 ≤ 𝑥 ≤ 1,0 ≤ 𝑦 ≤ 2. The region
1 1
where 𝑌 > 𝑋excludes only the triangle where 𝑌 ≤ 𝑋. The area of the triangle is 2 ⋅ 1 ⋅ 1 = 2.
2−1/2 3/2 3
The rectangle area is 2. Probability = = = 4.
2 2
Question 5
1
Are 𝑋and 𝑌independent if 𝑓(𝑥, 𝑦) = 2for 0 ≤ 𝑥 ≤ 1,0 ≤ 𝑦 ≤ 2?
,A. Yes
B. No
Rationale: The joint p.d.f. factors as 𝑓(𝑥, 𝑦) = 𝑓𝑋 (𝑥) ⋅ 𝑓𝑌 (𝑦)where 𝑓𝑋 (𝑥) = 1for 0 ≤ 𝑥 ≤
1
1and 𝑓𝑌 (𝑦) = 2for 0 ≤ 𝑦 ≤ 2. Since the domain is a rectangle and the joint density factors into
functions of 𝑥and 𝑦separately, 𝑋and 𝑌are independent .
Question 6
If 𝑋and 𝑌are i.i.d. Nor(1, 4) random variables, find Var(3X - 2Y).
A. 0
B. 10
C. 20
D. 52
E. 208
Rationale: Var(3X - 2Y) = 9Var(X) + 4Var(Y) - 12Cov(X,Y). Since X and Y are independent,
Cov(X,Y) = 0. Var(X) = Var(Y) = 4. Thus Var = 9(4) + 4(4) = 36 + 16 = 52 .
Question 7
If 𝑋and 𝑌are i.i.d. Unif(0,1) random variables, find E[2XY].
A. 0
B. 1/2
C. 1/4
D. 1
Rationale: E[2XY] = 2E[X]E[Y] by independence. E[X] = E[Y] = 1/2. Thus E[2XY] = 2(1/2)(1/2) =
1/2 .
Question 8
The covariance of 𝑋and 𝑌is 1/2. What can be concluded?
A. X and Y are positively correlated but could be independent
B. X and Y cannot be independent
, C. X and Y must be independent
D. X and Y have correlation coefficient 1
Rationale: Independence implies zero covariance. Since Cov(X,Y) = 1/2 ≠ 0, X and Y cannot
be independent. Non-zero covariance indicates a linear relationship exists .
Question 9
If 𝑈is Unif(0,1), how can we simulate a Geom(0.6) random variate?
A. [6𝑈] + 1
B. [ln(𝑈)/ln(0.4)]
C. [ln(𝑈)/ln(0.6)]
D. [𝑈/0.6]
Rationale: For geometric distribution with success probability p = 0.6, the inverse transform
method gives 𝑋 = ⌈ln(𝑈)/ln(1 − 𝑝)⌉or equivalently [ln(𝑈)/ln(0.4)](using floor). The
geometric is the number of trials until first success .
Question 10
Suppose 𝑈and 𝑉are PRNs. Let 𝑋 = 𝑈 + 𝑉. As the sample size increases, what p.d.f. does the
histogram of 𝑋approach?
A. Uniform
B. Normal
C. Triangular
D. Exponential
Rationale: The sum of two independent Unif(0,1) random variables follows a symmetric
triangular distribution on [0,2] with mode at 1. This is a classic convolution result .
Question 11 [SATA]
Which of the following are true about acceptance-rejection (A-R) generation? [Select all that
apply]
A. The closer the majorizing function 𝑡(𝑥)is to 𝑓(𝑥), the more efficient A-R is
B. ℎ(𝑦) = 𝑡(𝑦)/ ∫ 𝑡 (𝑥)𝑑𝑥must be a valid p.d.f.
