ISYE 6644 - FINAL PREP 2024
UPDATED QUESTIONS WITH
ANSWERS VERIFIED BY EXPERTS
Suppose that we want to know which of Coke, Pepsi, and Dr. Pepper is the most
popular. We would like to make the correct selection with probability of at least P ⋆ =
0.90 in the event that the ratio of the highest-to-second-highest preference probabilities
happens to be at least θ ⋆ = 1.4. If we use procedure M B E M, then the corresponding
table in the notes (with k = 3) tells us to take 126 samples (taste tests). Suppose we
take those samples sequentially and after 100 have been taken it turns out that 65
people prefer Coke, 25 love Pepsi, and 10 like Dr. Pepper. What to do? - CORRECT
ANSWER>>>>Stop the test now and declare with confidence of at least 90% that Coke
is the most-preferred.
Which of the following problems might best be characterized by a finite-horizon
simulation? - CORRECT ANSWER>>>>Simulating the operations of a bank from 9:00
a.m. until 5:00 p.m.
Let's run a simulation whose output is a sequence of daily inventory levels for a
particular product. Which of the following statements is true? - CORRECT
ANSWER>>>>The consecutive daily inventory levels may not be identically distributed.
Suppose that X 1 , X 2 , ... is a stationary (steady-state) stochastic process with
covariance function R k ≡ C o v ( X 1 , X 1 + k ), for k = 0 , 1 , .... We know from class
that the variance of the sample mean can be represented asV a r ( X ¯ n ) = 1 n [ R 0 +
2 ∑ k = 1 n − 1 ( 1 − k n ) R k ] .We also know from class that for a simple AR(1)
process, we have R k = ϕ k, k = 0 , 1 , 2 , ... Compute V a r ( X ¯ n ) for an AR(1)
process with n = 3 and ϕ = 0.8. - CORRECT ANSWER>>>>0.831
Suppose we want to estimate the expected average waiting time for the first m = 100
customers at a bank. We make r = 4 independent replications of the system, each
initialized empty and idle and consisting of 100 waiting times. The resulting replicate
means are:
i 1 2 3 4 Z i 5.2 4.3 3.1 4.2
Find a 90% confidence interval for the mean average waiting time for the first 100
customers. - CORRECT ANSWER>>>>[3.188,5.212]
Consider a particular data set of 100,000 stationary waiting times obtained from a large
queueing system. Suppose your goal is to get a confidence interval for the unknown
mean. Would you rather use (a) 50 batches of 2000 observations or (b) 10000 batches
, of 10 observations each? - CORRECT ANSWER>>>>50 batches of 2000 observations
because the method of batch means requires a very large batch size
Consider the output analysis method of non overlapping batch means. Assuming that
you have a sufficiently large batch size, it can be shown that when the number of
batches b is even, the expected width of the 90% two-sided confidence interval for μ is
proportional tot 0.05 , b − 1 b − 1 ( b − 1 2 ) ( b − 3 2 ) ⋯ 1 2 ( b − 2 2 ) ! .Using the
above equation, determine which of the following values of b gives the smallest
expected width. - CORRECT ANSWER>>>>b=6
Let h ( b ) denote the value of the above expression as a function of b. Then easy
calculations reveal that h ( b ) = 3.157, h ( 4 ) = 1.019, and h ( 6 ) = 0.845. So the
answer is b = 6
Consider the following observations:
54 70 75 62
If we choose a batch size of 3, calculate all of the overlapping batch means for me. -
CORRECT ANSWER>>>>66.3, 69.0
X1,3 = 1/3 ΣXi = 66.3 and X2,3
= 1/3 ΣXi = 69.
TRUE or FALSE? Simulation output (e.g., consecutive customer waiting times) is
almost never i.i.d. normal - and that's a big fat problem - CORRECT
ANSWER>>>>TRUE
We often distinguish between two general types of simulations with regard to output
analysis. What are they called? - CORRECT ANSWER>>>>Finite-horizon and
steadystate
TRUE or FALSE? Suppose that X 1 , X 2 , . . . , X n are consecutive waiting times, and
we define the sample mean X ¯ = ∑ i = 1 n X i / n. Then V a r ( X ¯ ) = V a r ( X i ) / n. -
CORRECT ANSWER>>>>Very FALSE! (The issue is that correlation between the
observations messes up the variance of the sample mean. In fact, this is one of the
main reasons why output analysis is difficult!)
Which of the following scenarios might be well-suited for a finite-horizon analysis? -
CORRECT 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? The main method of attack for terminating simulations is via
independent replications. - CORRECT ANSWER>>>>TRUE (Independent replications
help to mitigate issues with simulation outputs not being i.i.d.)
UPDATED QUESTIONS WITH
ANSWERS VERIFIED BY EXPERTS
Suppose that we want to know which of Coke, Pepsi, and Dr. Pepper is the most
popular. We would like to make the correct selection with probability of at least P ⋆ =
0.90 in the event that the ratio of the highest-to-second-highest preference probabilities
happens to be at least θ ⋆ = 1.4. If we use procedure M B E M, then the corresponding
table in the notes (with k = 3) tells us to take 126 samples (taste tests). Suppose we
take those samples sequentially and after 100 have been taken it turns out that 65
people prefer Coke, 25 love Pepsi, and 10 like Dr. Pepper. What to do? - CORRECT
ANSWER>>>>Stop the test now and declare with confidence of at least 90% that Coke
is the most-preferred.
Which of the following problems might best be characterized by a finite-horizon
simulation? - CORRECT ANSWER>>>>Simulating the operations of a bank from 9:00
a.m. until 5:00 p.m.
Let's run a simulation whose output is a sequence of daily inventory levels for a
particular product. Which of the following statements is true? - CORRECT
ANSWER>>>>The consecutive daily inventory levels may not be identically distributed.
Suppose that X 1 , X 2 , ... is a stationary (steady-state) stochastic process with
covariance function R k ≡ C o v ( X 1 , X 1 + k ), for k = 0 , 1 , .... We know from class
that the variance of the sample mean can be represented asV a r ( X ¯ n ) = 1 n [ R 0 +
2 ∑ k = 1 n − 1 ( 1 − k n ) R k ] .We also know from class that for a simple AR(1)
process, we have R k = ϕ k, k = 0 , 1 , 2 , ... Compute V a r ( X ¯ n ) for an AR(1)
process with n = 3 and ϕ = 0.8. - CORRECT ANSWER>>>>0.831
Suppose we want to estimate the expected average waiting time for the first m = 100
customers at a bank. We make r = 4 independent replications of the system, each
initialized empty and idle and consisting of 100 waiting times. The resulting replicate
means are:
i 1 2 3 4 Z i 5.2 4.3 3.1 4.2
Find a 90% confidence interval for the mean average waiting time for the first 100
customers. - CORRECT ANSWER>>>>[3.188,5.212]
Consider a particular data set of 100,000 stationary waiting times obtained from a large
queueing system. Suppose your goal is to get a confidence interval for the unknown
mean. Would you rather use (a) 50 batches of 2000 observations or (b) 10000 batches
, of 10 observations each? - CORRECT ANSWER>>>>50 batches of 2000 observations
because the method of batch means requires a very large batch size
Consider the output analysis method of non overlapping batch means. Assuming that
you have a sufficiently large batch size, it can be shown that when the number of
batches b is even, the expected width of the 90% two-sided confidence interval for μ is
proportional tot 0.05 , b − 1 b − 1 ( b − 1 2 ) ( b − 3 2 ) ⋯ 1 2 ( b − 2 2 ) ! .Using the
above equation, determine which of the following values of b gives the smallest
expected width. - CORRECT ANSWER>>>>b=6
Let h ( b ) denote the value of the above expression as a function of b. Then easy
calculations reveal that h ( b ) = 3.157, h ( 4 ) = 1.019, and h ( 6 ) = 0.845. So the
answer is b = 6
Consider the following observations:
54 70 75 62
If we choose a batch size of 3, calculate all of the overlapping batch means for me. -
CORRECT ANSWER>>>>66.3, 69.0
X1,3 = 1/3 ΣXi = 66.3 and X2,3
= 1/3 ΣXi = 69.
TRUE or FALSE? Simulation output (e.g., consecutive customer waiting times) is
almost never i.i.d. normal - and that's a big fat problem - CORRECT
ANSWER>>>>TRUE
We often distinguish between two general types of simulations with regard to output
analysis. What are they called? - CORRECT ANSWER>>>>Finite-horizon and
steadystate
TRUE or FALSE? Suppose that X 1 , X 2 , . . . , X n are consecutive waiting times, and
we define the sample mean X ¯ = ∑ i = 1 n X i / n. Then V a r ( X ¯ ) = V a r ( X i ) / n. -
CORRECT ANSWER>>>>Very FALSE! (The issue is that correlation between the
observations messes up the variance of the sample mean. In fact, this is one of the
main reasons why output analysis is difficult!)
Which of the following scenarios might be well-suited for a finite-horizon analysis? -
CORRECT 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? The main method of attack for terminating simulations is via
independent replications. - CORRECT ANSWER>>>>TRUE (Independent replications
help to mitigate issues with simulation outputs not being i.i.d.)
