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ISYE 6644 EXAM PREMIUM PRACTICE QUESTION BANK
ENGINEERING AND SCIENCE ## GEORGIA INSTITUTE
OF TECHNOLOGY | INDUSTRIAL & SYSTEMS
100% VERIFIED SOLUTIONS | UPDATED PER LATEST
## TABLE OF CONTENTS
| Section | Topic | Questions | Bloom's Levels Covered |
|---------|-------|-----------|------------------------|
| **Section 1** | Foundational Probability & Statistics Review | Q1–Q20 | Remember, Understand, Apply |
| **Section 2** | Introduction to Simulation & Modeling Paradigms | Q21–Q40 | Understand, Apply, Analyze |
| **Section 3** | Random Number Generation & Statistical Testing | Q41–Q65 | Apply, Analyze, Evaluate |
| **Section 4** | Random Variate Generation Techniques | Q66–Q90 | Apply, Analyze, Create |
| **Section 5** | Input Modeling & Distribution Fitting | Q91–Q110 | Apply, Analyze, Evaluate |
| **Section 6** | Output Analysis for a Single System | Q111–Q135 | Analyze, Evaluate, Create |
| **Section 7** | Variance Reduction & Comparison of Systems | Q136–Q155 | Analyze, Evaluate, Create |
| **Section 8** | Arena/Simio Simulation Language Concepts | Q156–Q175 | Remember, Understand, Apply |
| **Section 9** | Queueing Theory & Discrete-Event Systems | Q176–Q195 | Apply, Analyze, Evaluate |
| **Section 10** | Monte Carlo Methods & Advanced Applications | Q196–Q215 | Apply, Analyze, Create |
| **Section 11** | Comprehensive Final Exam Simulation | Q216–Q225 | All Levels |
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# SECTION 1: FOUNDATIONAL PROBABILITY & STATISTICS REVIEW
## Question 1
**The probability of any event A, denoted P(A), must always fall within which range?**
A. \(-1 \leq P(A) \leq 1\)
B. \(0 \leq P(A) \leq 1\)
C. \(0 < P(A) < 1\)
D. \(-\infty < P(A) < \infty\)
**CorreCt Answer: B**
**Rationale:** By the axioms of probability (Kolmogorov axioms), the probability of
any event is a non-negative real number, and the probability of the entire sample space is 1.
Therefore, \(0 \leq P(A) \leq 1\) for any event A. Option A is incorrect because probabilities
cannot be negative. Option C excludes the boundary values 0 and 1, which are valid probabilities
for impossible and certain events, respectively. Option D allows negative and unbounded values,
which violate probability axioms. This foundational concept is essential for all subsequent
probability calculations in simulation modeling.
---
, Page 3 of 177
## Question 2
**If X is a continuous random variable with cumulative distribution function (c.d.f.) F(x), what
is the distribution of the random variable F(X)?**
A. Normal(0,1)
B. Uniform(0,1)
C. Exponential(1)
D. Beta(1,1)
**CorreCt Answer: B**
**Rationale:** This is the **probability integral transform theorem**. For any
continuous random variable X with c.d.f. F(x), the random variable U = F(X) follows a
Uniform(0,1) distribution. This is because \(P(F(X) \leq u) = P(X \leq F^{-1}(u)) = F(F^{-1}(u))
= u\) for \(0 \leq u \leq 1\). This theorem is the theoretical foundation of the inverse transform
method for random variate generation, which is a core technique in ISYE 6644. Option D
(Beta(1,1)) is mathematically equivalent to Uniform(0,1), but B is the standard and expected
answer.
---
## Question 3
**Suppose X ~ Exponential(λ) with c.d.f. \(F(x) = 1 - e^{-\lambda x}\). What is the distribution
of the random variable \(1 - e^{-\lambda X}\)?**
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A. Exponential(λ)
B. Normal(0,1)
C. Uniform(0,1)
D. Gamma(1, λ)
**CorreCt Answer: C**
**Rationale:** By the probability integral transform theorem, since \(F(X) = 1 - e^{-
\lambda X}\) is the c.d.f. of the exponential distribution evaluated at X, the random variable
\(F(X) = 1 - e^{-\lambda X}\) follows Uniform(0,1). This property allows us to generate
exponential random variates from uniform random numbers: if U ~ Uniform(0,1), then \(X = -
\frac{1}{\lambda}\ln(1-U)\) is Exponential(λ). This is the inverse transform method applied to
the exponential distribution.
---
## Question 4
**If U is a Uniform(0,1) random variable, what is the distribution of \(-\frac{1}{\lambda}\ln(1-
U)\)?**
A. Uniform(0,1)
B. Normal(0,1)
ISYE 6644 EXAM PREMIUM PRACTICE QUESTION BANK
ENGINEERING AND SCIENCE ## GEORGIA INSTITUTE
OF TECHNOLOGY | INDUSTRIAL & SYSTEMS
100% VERIFIED SOLUTIONS | UPDATED PER LATEST
## TABLE OF CONTENTS
| Section | Topic | Questions | Bloom's Levels Covered |
|---------|-------|-----------|------------------------|
| **Section 1** | Foundational Probability & Statistics Review | Q1–Q20 | Remember, Understand, Apply |
| **Section 2** | Introduction to Simulation & Modeling Paradigms | Q21–Q40 | Understand, Apply, Analyze |
| **Section 3** | Random Number Generation & Statistical Testing | Q41–Q65 | Apply, Analyze, Evaluate |
| **Section 4** | Random Variate Generation Techniques | Q66–Q90 | Apply, Analyze, Create |
| **Section 5** | Input Modeling & Distribution Fitting | Q91–Q110 | Apply, Analyze, Evaluate |
| **Section 6** | Output Analysis for a Single System | Q111–Q135 | Analyze, Evaluate, Create |
| **Section 7** | Variance Reduction & Comparison of Systems | Q136–Q155 | Analyze, Evaluate, Create |
| **Section 8** | Arena/Simio Simulation Language Concepts | Q156–Q175 | Remember, Understand, Apply |
| **Section 9** | Queueing Theory & Discrete-Event Systems | Q176–Q195 | Apply, Analyze, Evaluate |
| **Section 10** | Monte Carlo Methods & Advanced Applications | Q196–Q215 | Apply, Analyze, Create |
| **Section 11** | Comprehensive Final Exam Simulation | Q216–Q225 | All Levels |
, Page 2 of 177
# SECTION 1: FOUNDATIONAL PROBABILITY & STATISTICS REVIEW
## Question 1
**The probability of any event A, denoted P(A), must always fall within which range?**
A. \(-1 \leq P(A) \leq 1\)
B. \(0 \leq P(A) \leq 1\)
C. \(0 < P(A) < 1\)
D. \(-\infty < P(A) < \infty\)
**CorreCt Answer: B**
**Rationale:** By the axioms of probability (Kolmogorov axioms), the probability of
any event is a non-negative real number, and the probability of the entire sample space is 1.
Therefore, \(0 \leq P(A) \leq 1\) for any event A. Option A is incorrect because probabilities
cannot be negative. Option C excludes the boundary values 0 and 1, which are valid probabilities
for impossible and certain events, respectively. Option D allows negative and unbounded values,
which violate probability axioms. This foundational concept is essential for all subsequent
probability calculations in simulation modeling.
---
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## Question 2
**If X is a continuous random variable with cumulative distribution function (c.d.f.) F(x), what
is the distribution of the random variable F(X)?**
A. Normal(0,1)
B. Uniform(0,1)
C. Exponential(1)
D. Beta(1,1)
**CorreCt Answer: B**
**Rationale:** This is the **probability integral transform theorem**. For any
continuous random variable X with c.d.f. F(x), the random variable U = F(X) follows a
Uniform(0,1) distribution. This is because \(P(F(X) \leq u) = P(X \leq F^{-1}(u)) = F(F^{-1}(u))
= u\) for \(0 \leq u \leq 1\). This theorem is the theoretical foundation of the inverse transform
method for random variate generation, which is a core technique in ISYE 6644. Option D
(Beta(1,1)) is mathematically equivalent to Uniform(0,1), but B is the standard and expected
answer.
---
## Question 3
**Suppose X ~ Exponential(λ) with c.d.f. \(F(x) = 1 - e^{-\lambda x}\). What is the distribution
of the random variable \(1 - e^{-\lambda X}\)?**
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A. Exponential(λ)
B. Normal(0,1)
C. Uniform(0,1)
D. Gamma(1, λ)
**CorreCt Answer: C**
**Rationale:** By the probability integral transform theorem, since \(F(X) = 1 - e^{-
\lambda X}\) is the c.d.f. of the exponential distribution evaluated at X, the random variable
\(F(X) = 1 - e^{-\lambda X}\) follows Uniform(0,1). This property allows us to generate
exponential random variates from uniform random numbers: if U ~ Uniform(0,1), then \(X = -
\frac{1}{\lambda}\ln(1-U)\) is Exponential(λ). This is the inverse transform method applied to
the exponential distribution.
---
## Question 4
**If U is a Uniform(0,1) random variable, what is the distribution of \(-\frac{1}{\lambda}\ln(1-
U)\)?**
A. Uniform(0,1)
B. Normal(0,1)
