ISYE 6644 - GT FINAL TEST 2026
QUESTIONS WITH CORRECT ANSWERS
GRADED A+
◍ Suppose that estimator A has bias = 3 and variance = 12, while estimator B
has bias -2 and variance = 14. Which estimator (A or B) has the lower mean
squared error?.
Answer: MSE = Bias^ 2 + Var, soM S E ( A ) = 9 + 12 = 21 and M S E ( B )
= 4 + 14 = 18.
◍ Which of the following scenarios might be well-suited for a finite-horizon
analysis?.
Answer: Simulate bank operations from 8:00 a.m. to 5:00 p.m.Simulate an
inventory system until the first stock-out occurs
◍ TRUE or FALSE? If you can't find a good theoretical distribution to model
a certain random variable, you might want to use the empirical distribution
of the data to do so..
Answer: TRUE. That's the point of this lesson!
◍ Consider the waiting time W i Q of the i th customer in an M / M / 1
queuing system with first-in-first-out services. What relation allows you to
calculate customer ( i + 1 )'s waiting time W i + 1 Q based on W i Q,
customer i's service time S i, and customer ( i + 1 )'s interarrival time I i +
1?.
Answer: Lindley's equationW^Q_{i+1} = max\{0, W^Q_i + S_i - I_{i+1}\}
◍ Suppose that we have a number of observations from a Pois( λ ) distribution,
and it turns out that the MLE for λ is λ ^ = 5. What's the maximum
likelihood estimate of Pr ( X = 3 )?.
Answer: By invariance,Pr ^ ( X = k ) = e − λ ^ λ ^ k k ! ,so that we have Pr ^
, ( X = 3 ) = e − x ¯ x ¯ k k ! = e − 5 5 3 3 ! = 0.1404.
◍ TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g.,
for the N o r ( μ , σ 2 ) distribution..
Answer: TRUE (it's possible based on taking the partial derivatives of the
likelihood function with respect to each parameter).
◍ Brownian motion is also known as ....
Answer: Wiener process
◍ Consider the constant c = ∫ R t ( x ) d x = 5. On average, how many
iterations (trials) will the A-R algorithm require?.
Answer: 5
◍ Which of the following problematic issues can arise in input data analysis?.
Answer: Not enough dataData coming from strange-looking, "non-standard"
distributionsNonstationary data (in which the distribution appears to change
over time)Correlated data
◍ What does GIGO mean?.
Answer: Garbage-in-garbage-out
◍ If X 1 , ... , X 10 are i.i.d. Pois(6), what is the expected value of the sample
variance S^2?.
Answer: 6S^2 is always unbiased for the variance of X i. Thus, we have E [
S^2 ] = V a r ( X i ) = λ = 6
◍ Name That Distribution!IQs.
Answer: Normal
◍ If is Unif(0,1), what is 6 ? (Note that . is the round-up function.).
Answer: A 6-side die toss
◍ Random Variate Generation.
Answer: The process of generating random numbers that follow a specific
probability distribution, based on Unif(0,1) pseudo-random numbers.
◍ TRUE or FALSE? You can find the inverse c.d.f. Φ^−1(. ) of the standard
normal distribution in closed form..
, Answer: FALSE. You need to use an approximation.
◍ Consider the PRN's U 1 = 0.1 , U 2 = 0.9 , and U 3 = 0.2. Use
Kolmogorov-Smirnov with α = 0.05 to test to see if these numbers are
indeed uniform. Do we ACCEPT or REJECT uniformity?.
Answer: AcceptSince D < D α , n, we ACCEPT uniformity (though it's kind
of a joke since it's only based on 3 observations)
◍ Variance Reduction Techniques.
Answer: Methods used to improve the efficiency of a simulation by reducing
the variability of the output statistics.
◍ Consider four observations from some unknown distribution, X 1 = 1.5, X 2
= − 3.7, X 3 = 2.7, and X 4 = 0.6. What is the fourth order statistic, which
we denoted by X ( 4 ) in class?.
Answer: 2.7merely means the largest of the sample of 4 observations.
◍ If U is Unif(0,1), how can we simulate a Geom(0.6) random variate?.
Answer: ln ( U ) / ln ( 0.4 ) ln ( 1 − U ) / ln ( 0.4 )
◍ Name That Distribution!The number of dice tosses until a 3 comes up..
Answer: Geometric
◍ Verification.
Answer: The process of ensuring that the simulation model is implemented
correctly and that the program runs as intended.
◍ TRUE or FALSE? The paired CI for the differences in two means is
designed to work especially well if all of the observations from the first
population are completely independent of all of the observations from the
second population..
Answer: FALSE. {In fact, it's easier to distinguish between the two means if
Xi is positively correlated with Yi. Think about my parallel parking example
in the class notes.}
◍ If X is a Nor(0,1) random variate, and Φ ( x ) is the Nor(0,1) c.d.f., what is
the distribution of Φ ( X )?.
QUESTIONS WITH CORRECT ANSWERS
GRADED A+
◍ Suppose that estimator A has bias = 3 and variance = 12, while estimator B
has bias -2 and variance = 14. Which estimator (A or B) has the lower mean
squared error?.
Answer: MSE = Bias^ 2 + Var, soM S E ( A ) = 9 + 12 = 21 and M S E ( B )
= 4 + 14 = 18.
◍ Which of the following scenarios might be well-suited for a finite-horizon
analysis?.
Answer: Simulate bank operations from 8:00 a.m. to 5:00 p.m.Simulate an
inventory system until the first stock-out occurs
◍ TRUE or FALSE? If you can't find a good theoretical distribution to model
a certain random variable, you might want to use the empirical distribution
of the data to do so..
Answer: TRUE. That's the point of this lesson!
◍ Consider the waiting time W i Q of the i th customer in an M / M / 1
queuing system with first-in-first-out services. What relation allows you to
calculate customer ( i + 1 )'s waiting time W i + 1 Q based on W i Q,
customer i's service time S i, and customer ( i + 1 )'s interarrival time I i +
1?.
Answer: Lindley's equationW^Q_{i+1} = max\{0, W^Q_i + S_i - I_{i+1}\}
◍ Suppose that we have a number of observations from a Pois( λ ) distribution,
and it turns out that the MLE for λ is λ ^ = 5. What's the maximum
likelihood estimate of Pr ( X = 3 )?.
Answer: By invariance,Pr ^ ( X = k ) = e − λ ^ λ ^ k k ! ,so that we have Pr ^
, ( X = 3 ) = e − x ¯ x ¯ k k ! = e − 5 5 3 3 ! = 0.1404.
◍ TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g.,
for the N o r ( μ , σ 2 ) distribution..
Answer: TRUE (it's possible based on taking the partial derivatives of the
likelihood function with respect to each parameter).
◍ Brownian motion is also known as ....
Answer: Wiener process
◍ Consider the constant c = ∫ R t ( x ) d x = 5. On average, how many
iterations (trials) will the A-R algorithm require?.
Answer: 5
◍ Which of the following problematic issues can arise in input data analysis?.
Answer: Not enough dataData coming from strange-looking, "non-standard"
distributionsNonstationary data (in which the distribution appears to change
over time)Correlated data
◍ What does GIGO mean?.
Answer: Garbage-in-garbage-out
◍ If X 1 , ... , X 10 are i.i.d. Pois(6), what is the expected value of the sample
variance S^2?.
Answer: 6S^2 is always unbiased for the variance of X i. Thus, we have E [
S^2 ] = V a r ( X i ) = λ = 6
◍ Name That Distribution!IQs.
Answer: Normal
◍ If is Unif(0,1), what is 6 ? (Note that . is the round-up function.).
Answer: A 6-side die toss
◍ Random Variate Generation.
Answer: The process of generating random numbers that follow a specific
probability distribution, based on Unif(0,1) pseudo-random numbers.
◍ TRUE or FALSE? You can find the inverse c.d.f. Φ^−1(. ) of the standard
normal distribution in closed form..
, Answer: FALSE. You need to use an approximation.
◍ Consider the PRN's U 1 = 0.1 , U 2 = 0.9 , and U 3 = 0.2. Use
Kolmogorov-Smirnov with α = 0.05 to test to see if these numbers are
indeed uniform. Do we ACCEPT or REJECT uniformity?.
Answer: AcceptSince D < D α , n, we ACCEPT uniformity (though it's kind
of a joke since it's only based on 3 observations)
◍ Variance Reduction Techniques.
Answer: Methods used to improve the efficiency of a simulation by reducing
the variability of the output statistics.
◍ Consider four observations from some unknown distribution, X 1 = 1.5, X 2
= − 3.7, X 3 = 2.7, and X 4 = 0.6. What is the fourth order statistic, which
we denoted by X ( 4 ) in class?.
Answer: 2.7merely means the largest of the sample of 4 observations.
◍ If U is Unif(0,1), how can we simulate a Geom(0.6) random variate?.
Answer: ln ( U ) / ln ( 0.4 ) ln ( 1 − U ) / ln ( 0.4 )
◍ Name That Distribution!The number of dice tosses until a 3 comes up..
Answer: Geometric
◍ Verification.
Answer: The process of ensuring that the simulation model is implemented
correctly and that the program runs as intended.
◍ TRUE or FALSE? The paired CI for the differences in two means is
designed to work especially well if all of the observations from the first
population are completely independent of all of the observations from the
second population..
Answer: FALSE. {In fact, it's easier to distinguish between the two means if
Xi is positively correlated with Yi. Think about my parallel parking example
in the class notes.}
◍ If X is a Nor(0,1) random variate, and Φ ( x ) is the Nor(0,1) c.d.f., what is
the distribution of Φ ( X )?.
