ISYE 6644 Exam 2 Simulation and Modeling
Georgia Tech Actual Exam 2026/2027 |
Complete Exam-Style Questions with
Detailed Rationales | Pass Guaranteed – A+
Graded
**SECTION 1: Random Number Generation & Statistical Testing (Questions 1–13)**
Q1: In ISYE 6644, the two most important statistical properties that the output of a good
pseudorandom number generator should appear to have are:
A. Normality and symmetry
B. Uniformity and independence [CORRECT]
C. Periodicity and monotonicity
D. Stationarity and ergodicity
Correct Answer: B
Rationale: Correct because a good pseudorandom generator produces a deterministic
sequence whose values appear to be independent and identically distributed Uniform(0,1), even
though the sequence is fully reproducible.
Q2: To check whether 10,000 generated random numbers are uniformly distributed over [0,1],
an analyst divides the interval into 10 equal subintervals and compares the observed count in
each subinterval with the expected count of 1,000. Which test is being applied?
A. Runs test
B. Kolmogorov–Smirnov test
C. Chi-square goodness-of-fit test [CORRECT]
D. Student's t test
Correct Answer: C
Rationale: Correct because the chi-square goodness-of-fit test compares observed cell
frequencies with expected frequencies under the hypothesized Uniform(0,1) distribution, making
it the standard test for uniformity.
Q3: A sequence of random numbers is examined for the lengths of its monotonically increasing
and decreasing subsequences. Which property of the generator does this evaluate?
A. Uniformity of the marginal distribution
B. Length of the period
C. Efficiency of the algorithm
D. Independence between successive random numbers [CORRECT]
Correct Answer: D
, Rationale: Correct because the runs test checks for correlation or dependence between
successive random numbers; too few or too many runs relative to the i.i.d. expectation signals a
lack of independence.
Q4: The Kolmogorov–Smirnov test assesses the uniformity of pseudorandom numbers by:
A. Measuring the maximum absolute difference between the empirical CDF of the sample and
the Uniform(0,1) CDF [CORRECT]
B. Comparing observed and expected counts in ten equal intervals
C. Counting sign changes in successive differences of the stream
D. Estimating the generator's cycle length
Correct Answer: A
Rationale: Correct because the K–S goodness-of-fit test is based on the supremum distance
between the empirical CDF and the theoretical CDF, and it is especially useful for continuous
distributions and small samples.
Q5: A linear congruential generator uses modulus m = 2^31 − 1 (a Mersenne prime) and
multiplier a = 7^5 = 16807, which is a primitive root modulo m. With a non-zero seed, the
maximum achievable cycle length is:
A. 2^31 − 1
B. 2^31
C. 2^31 − 2 [CORRECT]
D. 16807
Correct Answer: C
Rationale: Correct because a generator with prime modulus m and primitive-root multiplier
achieves full period m − 1; since a non-zero seed can never produce 0, the cycle length is 2^31
− 2.
Q6: The Box–Muller transform is a technique used to:
A. Convert pairs of independent Uniform(0,1) variates into pairs of independent standard normal
variates [CORRECT]
B. Convert uniform variates into independent exponential variates
C. Test a random number stream for independence
D. Extend the period of a linear congruential generator
Correct Answer: A
Rationale: Correct because Box–Muller applies the probability integral transform to a bivariate
uniform — Z1 = sqrt(−2 ln U1)cos(2πU2) and Z2 = sqrt(−2 ln U1)sin(2πU2) — producing
independent N(0,1) variates.
Q7: A newly proposed generator produces the stream 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, .... Although a
histogram of many values might look flat, this stream would most clearly fail which statistical
test?
A. Chi-square goodness-of-fit test
B. Kolmogorov–Smirnov test
C. Runs test [CORRECT]
Georgia Tech Actual Exam 2026/2027 |
Complete Exam-Style Questions with
Detailed Rationales | Pass Guaranteed – A+
Graded
**SECTION 1: Random Number Generation & Statistical Testing (Questions 1–13)**
Q1: In ISYE 6644, the two most important statistical properties that the output of a good
pseudorandom number generator should appear to have are:
A. Normality and symmetry
B. Uniformity and independence [CORRECT]
C. Periodicity and monotonicity
D. Stationarity and ergodicity
Correct Answer: B
Rationale: Correct because a good pseudorandom generator produces a deterministic
sequence whose values appear to be independent and identically distributed Uniform(0,1), even
though the sequence is fully reproducible.
Q2: To check whether 10,000 generated random numbers are uniformly distributed over [0,1],
an analyst divides the interval into 10 equal subintervals and compares the observed count in
each subinterval with the expected count of 1,000. Which test is being applied?
A. Runs test
B. Kolmogorov–Smirnov test
C. Chi-square goodness-of-fit test [CORRECT]
D. Student's t test
Correct Answer: C
Rationale: Correct because the chi-square goodness-of-fit test compares observed cell
frequencies with expected frequencies under the hypothesized Uniform(0,1) distribution, making
it the standard test for uniformity.
Q3: A sequence of random numbers is examined for the lengths of its monotonically increasing
and decreasing subsequences. Which property of the generator does this evaluate?
A. Uniformity of the marginal distribution
B. Length of the period
C. Efficiency of the algorithm
D. Independence between successive random numbers [CORRECT]
Correct Answer: D
, Rationale: Correct because the runs test checks for correlation or dependence between
successive random numbers; too few or too many runs relative to the i.i.d. expectation signals a
lack of independence.
Q4: The Kolmogorov–Smirnov test assesses the uniformity of pseudorandom numbers by:
A. Measuring the maximum absolute difference between the empirical CDF of the sample and
the Uniform(0,1) CDF [CORRECT]
B. Comparing observed and expected counts in ten equal intervals
C. Counting sign changes in successive differences of the stream
D. Estimating the generator's cycle length
Correct Answer: A
Rationale: Correct because the K–S goodness-of-fit test is based on the supremum distance
between the empirical CDF and the theoretical CDF, and it is especially useful for continuous
distributions and small samples.
Q5: A linear congruential generator uses modulus m = 2^31 − 1 (a Mersenne prime) and
multiplier a = 7^5 = 16807, which is a primitive root modulo m. With a non-zero seed, the
maximum achievable cycle length is:
A. 2^31 − 1
B. 2^31
C. 2^31 − 2 [CORRECT]
D. 16807
Correct Answer: C
Rationale: Correct because a generator with prime modulus m and primitive-root multiplier
achieves full period m − 1; since a non-zero seed can never produce 0, the cycle length is 2^31
− 2.
Q6: The Box–Muller transform is a technique used to:
A. Convert pairs of independent Uniform(0,1) variates into pairs of independent standard normal
variates [CORRECT]
B. Convert uniform variates into independent exponential variates
C. Test a random number stream for independence
D. Extend the period of a linear congruential generator
Correct Answer: A
Rationale: Correct because Box–Muller applies the probability integral transform to a bivariate
uniform — Z1 = sqrt(−2 ln U1)cos(2πU2) and Z2 = sqrt(−2 ln U1)sin(2πU2) — producing
independent N(0,1) variates.
Q7: A newly proposed generator produces the stream 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, .... Although a
histogram of many values might look flat, this stream would most clearly fail which statistical
test?
A. Chi-square goodness-of-fit test
B. Kolmogorov–Smirnov test
C. Runs test [CORRECT]
