ISYE-6644 EXAM 1 QUESTIONS AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS
RATIONALES 2026 Q&A | INSTANT DOWNLOAD
PDF
Core Domains
• Basic Probability and Conditional Probability
• Random Variables and Probability Distributions
• Discrete Distributions
• Continuous Distributions
• Expected Values, Variance, and Moments
• Joint, Marginal, and Conditional Distributions
• Covariance, Correlation, and Independence
• Law of Large Numbers and Central Limit Theorem
• Poisson Processes and Arrival Modeling
• Statistical Estimation and Inference Basics
Introduction
This examination assesses foundational and applied knowledge in
simulation and probabilistic modeling for engineering and science. It
evaluates understanding of probability axioms, random variables,
common discrete and continuous distributions, expected values,
variance, joint distributions, covariance, correlation, limit theorems, and
Poisson processes. The multiple-choice and scenario-based format
,emphasizes real-world application, critical thinking, and quantitative
decision-making. Candidates are tested on their ability to model
uncertainty, derive probabilistic quantities, and interpret results in
engineering and scientific contexts.
SECTION ONE: QUESTIONS 1–100
Question 1
If X is a Bernoulli random variable with success probability p, what is
Var(X)?
A. p
B. p²
C. p(1 − p)
D. 1 − p
C. p(1 − p)
RATIONALE: For a Bernoulli(p) random variable, E[X] = p and E[X²] =
p, so Var(X) = E[X²] − (E[X])² = p − p² = p(1 − p).
Question 2
In a Poisson process with rate λ, the interarrival times are independent
and identically distributed as which distribution?
A. Poisson(λ)
B. Exponential(λ)
C. Normal(1/λ, 1/λ²)
D. Uniform(0, λ)
B. Exponential(λ)
, RATIONALE: The interarrival times of a Poisson process with rate λ
are i.i.d. Exponential with mean 1/λ. The number of arrivals in a fixed
interval is Poisson(λt).
Question 3
For a continuous random variable X, which statement is always true
about its cumulative distribution function F(x)?
A. F(x) is strictly increasing.
B. F(x) is nondecreasing and right-continuous.
C. F(x) is always differentiable.
D. F(x) is symmetric about the mean.
B. F(x) is nondecreasing and right-continuous.
RATIONALE: A valid CDF must be nondecreasing, right-continuous,
with limits 0 at −∞ and 1 at +∞. It need not be strictly increasing,
differentiable, or symmetric.
Question 4
If X ~ Binomial(n, p), which expression gives E[X²]?
A. np
B. np(1 − p)
C. np(1 − p) + n²p²
D. n²p²
C. np(1 − p) + n²p²
RATIONALE: For binomial, Var(X) = np(1 − p) and E[X] = np. Since
Var(X) = E[X²] − (E[X])², E[X²] = np(1 − p) + n²p².
Question 5
, A geometric random variable counts the number of trials until the first
success. If success probability is p, what is P(X = k)?
A. p(1 − p)^k
B. (1 − p)^(k−1) p
C. p^k (1 − p)
D. (1 − p)^k
B. (1 − p)^(k−1) p
RATIONALE: For the number of trials until the first success, the first
k−1 trials must be failures and the k-th trial a success, giving P(X = k) = (1
− p)^(k−1) p.
Question 6
If X ~ Poisson(λ), what is Var(X)?
A. λ
B. λ²
C. √λ
D. 1/λ
A. λ
RATIONALE: For a Poisson distribution with rate λ, both the mean and
variance equal λ.
Question 7
For X ~ Uniform(a, b), what is E[X]?
A. (a + b)/2
B. (b − a)/2
ANSWERS (VERIFIED ANSWERS) PLUS
RATIONALES 2026 Q&A | INSTANT DOWNLOAD
Core Domains
• Basic Probability and Conditional Probability
• Random Variables and Probability Distributions
• Discrete Distributions
• Continuous Distributions
• Expected Values, Variance, and Moments
• Joint, Marginal, and Conditional Distributions
• Covariance, Correlation, and Independence
• Law of Large Numbers and Central Limit Theorem
• Poisson Processes and Arrival Modeling
• Statistical Estimation and Inference Basics
Introduction
This examination assesses foundational and applied knowledge in
simulation and probabilistic modeling for engineering and science. It
evaluates understanding of probability axioms, random variables,
common discrete and continuous distributions, expected values,
variance, joint distributions, covariance, correlation, limit theorems, and
Poisson processes. The multiple-choice and scenario-based format
,emphasizes real-world application, critical thinking, and quantitative
decision-making. Candidates are tested on their ability to model
uncertainty, derive probabilistic quantities, and interpret results in
engineering and scientific contexts.
SECTION ONE: QUESTIONS 1–100
Question 1
If X is a Bernoulli random variable with success probability p, what is
Var(X)?
A. p
B. p²
C. p(1 − p)
D. 1 − p
C. p(1 − p)
RATIONALE: For a Bernoulli(p) random variable, E[X] = p and E[X²] =
p, so Var(X) = E[X²] − (E[X])² = p − p² = p(1 − p).
Question 2
In a Poisson process with rate λ, the interarrival times are independent
and identically distributed as which distribution?
A. Poisson(λ)
B. Exponential(λ)
C. Normal(1/λ, 1/λ²)
D. Uniform(0, λ)
B. Exponential(λ)
, RATIONALE: The interarrival times of a Poisson process with rate λ
are i.i.d. Exponential with mean 1/λ. The number of arrivals in a fixed
interval is Poisson(λt).
Question 3
For a continuous random variable X, which statement is always true
about its cumulative distribution function F(x)?
A. F(x) is strictly increasing.
B. F(x) is nondecreasing and right-continuous.
C. F(x) is always differentiable.
D. F(x) is symmetric about the mean.
B. F(x) is nondecreasing and right-continuous.
RATIONALE: A valid CDF must be nondecreasing, right-continuous,
with limits 0 at −∞ and 1 at +∞. It need not be strictly increasing,
differentiable, or symmetric.
Question 4
If X ~ Binomial(n, p), which expression gives E[X²]?
A. np
B. np(1 − p)
C. np(1 − p) + n²p²
D. n²p²
C. np(1 − p) + n²p²
RATIONALE: For binomial, Var(X) = np(1 − p) and E[X] = np. Since
Var(X) = E[X²] − (E[X])², E[X²] = np(1 − p) + n²p².
Question 5
, A geometric random variable counts the number of trials until the first
success. If success probability is p, what is P(X = k)?
A. p(1 − p)^k
B. (1 − p)^(k−1) p
C. p^k (1 − p)
D. (1 − p)^k
B. (1 − p)^(k−1) p
RATIONALE: For the number of trials until the first success, the first
k−1 trials must be failures and the k-th trial a success, giving P(X = k) = (1
− p)^(k−1) p.
Question 6
If X ~ Poisson(λ), what is Var(X)?
A. λ
B. λ²
C. √λ
D. 1/λ
A. λ
RATIONALE: For a Poisson distribution with rate λ, both the mean and
variance equal λ.
Question 7
For X ~ Uniform(a, b), what is E[X]?
A. (a + b)/2
B. (b − a)/2
