ISYE 6644 OAN/O01/AO Final Exam
Simulation – Questions and Answers |
spring 2026 | 100% Correct – GT.
1. A linear congruential generator (LCG) uses modulus 𝑚 = 231 − 1, multiplier 𝑎 = 75 = 16807,
and increment 𝑐 = 0. What is its period?
A) 231 − 1
B) 231 − 2
C) 230
D) 231
Correct Answer: B
Rationale: With 𝑐 = 0, the LCG is multiplicative. For prime 𝑚, the maximum period is 𝑚 − 1 =
231 − 2 when the multiplier is a primitive root modulo 𝑚. The given 𝑎 = 16807 achieves this full
period.
2. Which of the following conditions guarantees a full period (𝑚) for a mixed LCG (𝑐 > 0)?
A) 𝑚 prime, 𝑎 primitive root
B) 𝑐 relatively prime to 𝑚, 𝑎 − 1 divisible by all prime factors of 𝑚, and if 4 ∣ 𝑚 then 4 ∣ 𝑎 − 1
C) 𝑎 and 𝑐 are both relatively prime to 𝑚
D) 𝑎 is a primitive root modulo 𝑚 and 𝑐 = 0
Correct Answer: B
Rationale: Hull-Dobell theorem for full period 𝑚 with 𝑐 ≠ 0 requires: 𝑐 and 𝑚 be relatively
prime, 𝑎 − 1 is a multiple of every prime dividing 𝑚, and 𝑎 − 1 is a multiple of 4 if 4 ∣ 𝑚.
,3. The inverse-transform method for generating a continuous random variate 𝑋 with
CDF 𝐹(𝑥) requires:
A) Solving 𝐹(𝑋) = 𝑈 for 𝑋
B) Generating 𝑋 = 𝐹 −1 (𝑈), where 𝑈 ∼ 𝑈(0, 1)
C) Computing 𝑋 = 𝐹(𝑈)
D) Both A and B are correct
Correct Answer: D
Rationale: Inverse-transform sets 𝑈 = 𝐹(𝑋), then inverts to 𝑋 = 𝐹 −1 (𝑈) for 𝑈 ∼ 𝑈(0, 1). Both
statements describe the same process.
4. To generate an exponential(𝜆) random variate using inverse transform, the formula is:
1
A) − 𝜆 ln(𝑈)
1
B) − 𝜆 ln(1 − 𝑈)
C) Both A and B are equivalent because 𝑈 and 1 − 𝑈 are both 𝑈(0, 1)
D) 𝜆ln(𝑈)
Correct Answer: C
1
Rationale: The exponential CDF inverse is − 𝜆 ln(1 − 𝑈). Since 1 − 𝑈 is also 𝑈(0, 1),
1
using − 𝜆 ln(𝑈) works identically.
5. The acceptance-rejection method requires:
A) A majorizing density 𝑡(𝑥) such that 𝑓(𝑥) ≤ 𝑐 ⋅ 𝑡(𝑥) for some constant 𝑐 ≥ 1
B) The ability to sample directly from 𝑓(𝑥)
C) 𝑡(𝑥) must be exactly proportional to 𝑓(𝑥)
D) 𝑓(𝑥) must be bounded on a finite interval
Correct Answer: A
,Rationale: Acceptance-rejection uses a proposal density 𝑡(𝑥) easy to sample, with 𝑓(𝑥) ≤ 𝑐 ⋅
𝑡(𝑥). Generate candidate from 𝑡(𝑥), accept with probability 𝑓(𝑥)/(𝑐 𝑡(𝑥)).
6. Using the Box-Muller method, two independent standard normal variates are generated from
two independent 𝑈(0, 1) variables 𝑈1 , 𝑈2 as:
A) 𝑍1 = √−2 ln 𝑈1 cos(2𝜋𝑈2 ), 𝑍2 = √−2 ln 𝑈1 sin(2𝜋𝑈2 )
B) 𝑍1 = √−2 ln 𝑈1 cos(𝜋𝑈2 ), 𝑍2 = √−2 ln 𝑈1 sin(𝜋𝑈2 )
C) 𝑍1 = −2ln(𝑈1 )cos(2𝜋𝑈2 ), 𝑍2 = −2ln(𝑈1 )sin(2𝜋𝑈2 )
D) 𝑍1 = ln(𝑈1 )cos(2𝜋𝑈2 ), 𝑍2 = ln(𝑈1 )sin(2𝜋𝑈2 )
Correct Answer: A
Rationale: Box-Muller transforms 𝑈1 , 𝑈2 via radius √−2 ln 𝑈1 and angle 2𝜋𝑈2 . This yields
independent 𝑁(0, 1) variates.
7. In simulation input modeling, a Q-Q plot is used to:
A) Test independence of data
B) Visually assess whether data come from a specified distribution
C) Compute autocorrelations
D) Determine the number of replications
Correct Answer: B
Rationale: A quantile-quantile plot compares sample quantiles to theoretical quantiles of a
candidate distribution; a straight line suggests good fit.
8. The Kolmogorov-Smirnov (KS) test for goodness-of-fit is:
A) Parametric test for means
, B) Applicable only to normal distributions
C) Based on the maximum absolute difference between the empirical CDF and the theoretical
CDF
D) Suitable only for discrete distributions
Correct Answer: C
Rationale: The KS statistic is 𝐷 = max𝑥 ∣ 𝐹𝑛 (𝑥) − 𝐹(𝑥) ∣, comparing the empirical and fitted
CDFs.
9. In a chi-square goodness-of-fit test, if the expected frequency in a bin is less than 5, one
should:
A) Ignore the bin
B) Double the bin width
C) Combine adjacent bins to increase expected counts
D) Use KS test instead because it has no such requirement
Correct Answer: C
Rationale: Chi-square approximation is poor for small expected frequencies. Combining bins
preserves test validity.
10. When estimating the mean from a terminating simulation, the method of independent
replications:
A) Uses a single long run with batch means
B) Makes multiple independent runs, each with its own initial conditions, and computes sample
mean and variance across runs
C) Requires a warm-up period deletion
D) Uses overlapping batches
Correct Answer: B
Simulation – Questions and Answers |
spring 2026 | 100% Correct – GT.
1. A linear congruential generator (LCG) uses modulus 𝑚 = 231 − 1, multiplier 𝑎 = 75 = 16807,
and increment 𝑐 = 0. What is its period?
A) 231 − 1
B) 231 − 2
C) 230
D) 231
Correct Answer: B
Rationale: With 𝑐 = 0, the LCG is multiplicative. For prime 𝑚, the maximum period is 𝑚 − 1 =
231 − 2 when the multiplier is a primitive root modulo 𝑚. The given 𝑎 = 16807 achieves this full
period.
2. Which of the following conditions guarantees a full period (𝑚) for a mixed LCG (𝑐 > 0)?
A) 𝑚 prime, 𝑎 primitive root
B) 𝑐 relatively prime to 𝑚, 𝑎 − 1 divisible by all prime factors of 𝑚, and if 4 ∣ 𝑚 then 4 ∣ 𝑎 − 1
C) 𝑎 and 𝑐 are both relatively prime to 𝑚
D) 𝑎 is a primitive root modulo 𝑚 and 𝑐 = 0
Correct Answer: B
Rationale: Hull-Dobell theorem for full period 𝑚 with 𝑐 ≠ 0 requires: 𝑐 and 𝑚 be relatively
prime, 𝑎 − 1 is a multiple of every prime dividing 𝑚, and 𝑎 − 1 is a multiple of 4 if 4 ∣ 𝑚.
,3. The inverse-transform method for generating a continuous random variate 𝑋 with
CDF 𝐹(𝑥) requires:
A) Solving 𝐹(𝑋) = 𝑈 for 𝑋
B) Generating 𝑋 = 𝐹 −1 (𝑈), where 𝑈 ∼ 𝑈(0, 1)
C) Computing 𝑋 = 𝐹(𝑈)
D) Both A and B are correct
Correct Answer: D
Rationale: Inverse-transform sets 𝑈 = 𝐹(𝑋), then inverts to 𝑋 = 𝐹 −1 (𝑈) for 𝑈 ∼ 𝑈(0, 1). Both
statements describe the same process.
4. To generate an exponential(𝜆) random variate using inverse transform, the formula is:
1
A) − 𝜆 ln(𝑈)
1
B) − 𝜆 ln(1 − 𝑈)
C) Both A and B are equivalent because 𝑈 and 1 − 𝑈 are both 𝑈(0, 1)
D) 𝜆ln(𝑈)
Correct Answer: C
1
Rationale: The exponential CDF inverse is − 𝜆 ln(1 − 𝑈). Since 1 − 𝑈 is also 𝑈(0, 1),
1
using − 𝜆 ln(𝑈) works identically.
5. The acceptance-rejection method requires:
A) A majorizing density 𝑡(𝑥) such that 𝑓(𝑥) ≤ 𝑐 ⋅ 𝑡(𝑥) for some constant 𝑐 ≥ 1
B) The ability to sample directly from 𝑓(𝑥)
C) 𝑡(𝑥) must be exactly proportional to 𝑓(𝑥)
D) 𝑓(𝑥) must be bounded on a finite interval
Correct Answer: A
,Rationale: Acceptance-rejection uses a proposal density 𝑡(𝑥) easy to sample, with 𝑓(𝑥) ≤ 𝑐 ⋅
𝑡(𝑥). Generate candidate from 𝑡(𝑥), accept with probability 𝑓(𝑥)/(𝑐 𝑡(𝑥)).
6. Using the Box-Muller method, two independent standard normal variates are generated from
two independent 𝑈(0, 1) variables 𝑈1 , 𝑈2 as:
A) 𝑍1 = √−2 ln 𝑈1 cos(2𝜋𝑈2 ), 𝑍2 = √−2 ln 𝑈1 sin(2𝜋𝑈2 )
B) 𝑍1 = √−2 ln 𝑈1 cos(𝜋𝑈2 ), 𝑍2 = √−2 ln 𝑈1 sin(𝜋𝑈2 )
C) 𝑍1 = −2ln(𝑈1 )cos(2𝜋𝑈2 ), 𝑍2 = −2ln(𝑈1 )sin(2𝜋𝑈2 )
D) 𝑍1 = ln(𝑈1 )cos(2𝜋𝑈2 ), 𝑍2 = ln(𝑈1 )sin(2𝜋𝑈2 )
Correct Answer: A
Rationale: Box-Muller transforms 𝑈1 , 𝑈2 via radius √−2 ln 𝑈1 and angle 2𝜋𝑈2 . This yields
independent 𝑁(0, 1) variates.
7. In simulation input modeling, a Q-Q plot is used to:
A) Test independence of data
B) Visually assess whether data come from a specified distribution
C) Compute autocorrelations
D) Determine the number of replications
Correct Answer: B
Rationale: A quantile-quantile plot compares sample quantiles to theoretical quantiles of a
candidate distribution; a straight line suggests good fit.
8. The Kolmogorov-Smirnov (KS) test for goodness-of-fit is:
A) Parametric test for means
, B) Applicable only to normal distributions
C) Based on the maximum absolute difference between the empirical CDF and the theoretical
CDF
D) Suitable only for discrete distributions
Correct Answer: C
Rationale: The KS statistic is 𝐷 = max𝑥 ∣ 𝐹𝑛 (𝑥) − 𝐹(𝑥) ∣, comparing the empirical and fitted
CDFs.
9. In a chi-square goodness-of-fit test, if the expected frequency in a bin is less than 5, one
should:
A) Ignore the bin
B) Double the bin width
C) Combine adjacent bins to increase expected counts
D) Use KS test instead because it has no such requirement
Correct Answer: C
Rationale: Chi-square approximation is poor for small expected frequencies. Combining bins
preserves test validity.
10. When estimating the mean from a terminating simulation, the method of independent
replications:
A) Uses a single long run with batch means
B) Makes multiple independent runs, each with its own initial conditions, and computes sample
mean and variance across runs
C) Requires a warm-up period deletion
D) Uses overlapping batches
Correct Answer: B
