ISYE 6644 Midterm Test 1 Actual Exam 2026/2027 – Complete
Exam-Style Questions with Detailed Rationales | 100%
Verified | Pass Guaranteed – A+ Graded
Section A: Probability & Statistics Review for Simulation (10
Questions)
Q1: In a discrete-event simulation of a manufacturing system, machine failure times
follow an exponential distribution with a mean of 50 hours. Given that a particular
machine has already operated for 30 hours without failure, what is the probability that it
will survive an additional 20 hours?
A. e^(-50/50) ≈ 0.368
B. e^(-20/80) ≈ 0.779
C. e^(-20/50) ≈ 0.670 [CORRECT]
D. 20/50 = 0.400
Correct Answer: C
Rationale: The exponential distribution is memoryless, so P(T > 50 | T > 30) = P(T > 20) =
e^(-20/50) ≈ 0.670. Option A incorrectly uses the total elapsed time in the exponent; B
incorrectly adds the conditioning time to the rate denominator; D confuses probability
with the ratio of times.
,Q2: Customers arrive at a service facility according to a Poisson process with rate λ = 4
per hour. What is the probability that exactly 2 customers arrive during a 30-minute
interval?
A. e^(-4) * ≈ 0.146
B. e^(-2) * ≈ 0.271 [CORRECT]
C. e^(-2) * 2 ≈ 0.271 (same value but wrong formula structure)
D. e^(-1) * ≈ 0.184
Correct Answer: B
Rationale: For a 30-minute interval, the Poisson mean is μ = 4 × 0.5 = 2. P(X = 2) = e^(-2)
* 2² / 2! = 2e^(-2) ≈ 0.271. Option A uses the hourly rate; C presents a numerically
equivalent but formulaically incorrect expression; D uses the wrong rate and factorial.
Q3: A simulation experiment generates n = 100 independent samples from a population
with mean μ = 10 and variance σ² = 25. By the Central Limit Theorem, the approximate
sampling distribution of the sample mean X̄ is:
A. N(10, 25)
B. N(10, 0.25) [CORRECT]
C. N(100, 25)
D. N(10, 5)
Correct Answer: B
, Rationale: The CLT states that X̄ is approximately normal with mean μ = 10 and variance
σ²/n = 25/100 = 0.25. Option A uses the population variance; C confuses sample size
with mean; D uses the standard deviation instead of variance.
Q4: A pilot simulation of 30 replications produces a 95% confidence interval for mean
system time of [4.2, 5.8] minutes. The analyst wants to reduce the interval width by half.
Approximately how many total replications are required?
A. 60
B. 90
C. 120 [CORRECT]
D. 240
Correct Answer: C
Rationale: Confidence interval width is inversely proportional to √n. To halve the width,
the sample size must increase by a factor of 4: 30 × 4 = 120. Option A doubles instead
of quadrupling; B uses an incorrect factor; D squares the factor.
Q5: Which of the following properties is mathematically equivalent to the memoryless
property of a continuous positive random variable?
A. The coefficient of variation equals 1
B. The hazard function is constant [CORRECT]
C. The skewness equals 2
D. The median equals the mean
Exam-Style Questions with Detailed Rationales | 100%
Verified | Pass Guaranteed – A+ Graded
Section A: Probability & Statistics Review for Simulation (10
Questions)
Q1: In a discrete-event simulation of a manufacturing system, machine failure times
follow an exponential distribution with a mean of 50 hours. Given that a particular
machine has already operated for 30 hours without failure, what is the probability that it
will survive an additional 20 hours?
A. e^(-50/50) ≈ 0.368
B. e^(-20/80) ≈ 0.779
C. e^(-20/50) ≈ 0.670 [CORRECT]
D. 20/50 = 0.400
Correct Answer: C
Rationale: The exponential distribution is memoryless, so P(T > 50 | T > 30) = P(T > 20) =
e^(-20/50) ≈ 0.670. Option A incorrectly uses the total elapsed time in the exponent; B
incorrectly adds the conditioning time to the rate denominator; D confuses probability
with the ratio of times.
,Q2: Customers arrive at a service facility according to a Poisson process with rate λ = 4
per hour. What is the probability that exactly 2 customers arrive during a 30-minute
interval?
A. e^(-4) * ≈ 0.146
B. e^(-2) * ≈ 0.271 [CORRECT]
C. e^(-2) * 2 ≈ 0.271 (same value but wrong formula structure)
D. e^(-1) * ≈ 0.184
Correct Answer: B
Rationale: For a 30-minute interval, the Poisson mean is μ = 4 × 0.5 = 2. P(X = 2) = e^(-2)
* 2² / 2! = 2e^(-2) ≈ 0.271. Option A uses the hourly rate; C presents a numerically
equivalent but formulaically incorrect expression; D uses the wrong rate and factorial.
Q3: A simulation experiment generates n = 100 independent samples from a population
with mean μ = 10 and variance σ² = 25. By the Central Limit Theorem, the approximate
sampling distribution of the sample mean X̄ is:
A. N(10, 25)
B. N(10, 0.25) [CORRECT]
C. N(100, 25)
D. N(10, 5)
Correct Answer: B
, Rationale: The CLT states that X̄ is approximately normal with mean μ = 10 and variance
σ²/n = 25/100 = 0.25. Option A uses the population variance; C confuses sample size
with mean; D uses the standard deviation instead of variance.
Q4: A pilot simulation of 30 replications produces a 95% confidence interval for mean
system time of [4.2, 5.8] minutes. The analyst wants to reduce the interval width by half.
Approximately how many total replications are required?
A. 60
B. 90
C. 120 [CORRECT]
D. 240
Correct Answer: C
Rationale: Confidence interval width is inversely proportional to √n. To halve the width,
the sample size must increase by a factor of 4: 30 × 4 = 120. Option A doubles instead
of quadrupling; B uses an incorrect factor; D squares the factor.
Q5: Which of the following properties is mathematically equivalent to the memoryless
property of a continuous positive random variable?
A. The coefficient of variation equals 1
B. The hazard function is constant [CORRECT]
C. The skewness equals 2
D. The median equals the mean
