ISYE 6644 COMPREHENSIVE QUESTIONS AND
ANSWERS SET A+
✔✔This is sort of the same as Question 2, except we have now used common random
numbers to induce positive correlation between the results of the two systems. Again
find a two-sided 95% CI for the difference in the means of the two systems. - ✔✔This is
a paired-t CI problem assuming unknown variance of the differences.
[-16.5, -3.5]
✔✔Suppose A and B are two identically distributed, unbiased, antithetic estimators for
the mean μ of some random variable, and let C = ( A + B ) / 2. Which of the following is
true? - ✔✔E [ C ] = μ and V a r ( C ) < V a r ( A ) / 2.
✔✔Suppose that you want to pick that one of three normal populations having the
largest mean. We'll assume that the variances of the three competitors are all known to
be equal to σ 2 = 4. (Ya, I know that this is a crazy, unrealistic assumption, but let's go
with it anyway, okey dokey?) I want to choose the best of the three populations with
probability of correct selection of 95% whenever the best population's mean happens to
be at least δ ⋆ = 1 larger than the second-best population's. How many observations
from each population does Bechhofer's procedure N B tell me to take before I can make
such a conclusion? - ✔✔Using the notation of the notes, we want to make sure to get
the right answer with probability of P ⋆ = 0.95 whenever μ [ k ] − μ [ k − 1 ] ≥ δ ⋆ = 1.
We simply go to NB's table with k = 3 and δ ⋆ / σ = to obtain a sample size of n =
30 from each population.
✔✔In the above problem, suppose that we take the necessary observations and we
come up with the following sample means: X ¯ 1 = 7.6, X ¯ 2 = 11.1, and X ¯ 3 = 3.6.
What do we do? - ✔✔Pick population 2 and say that we are right with probability at least
95%
✔✔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? - ✔✔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? - ✔✔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? - ✔✔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. - ✔✔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. - ✔✔[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? - ✔✔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. - ✔✔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
ANSWERS SET A+
✔✔This is sort of the same as Question 2, except we have now used common random
numbers to induce positive correlation between the results of the two systems. Again
find a two-sided 95% CI for the difference in the means of the two systems. - ✔✔This is
a paired-t CI problem assuming unknown variance of the differences.
[-16.5, -3.5]
✔✔Suppose A and B are two identically distributed, unbiased, antithetic estimators for
the mean μ of some random variable, and let C = ( A + B ) / 2. Which of the following is
true? - ✔✔E [ C ] = μ and V a r ( C ) < V a r ( A ) / 2.
✔✔Suppose that you want to pick that one of three normal populations having the
largest mean. We'll assume that the variances of the three competitors are all known to
be equal to σ 2 = 4. (Ya, I know that this is a crazy, unrealistic assumption, but let's go
with it anyway, okey dokey?) I want to choose the best of the three populations with
probability of correct selection of 95% whenever the best population's mean happens to
be at least δ ⋆ = 1 larger than the second-best population's. How many observations
from each population does Bechhofer's procedure N B tell me to take before I can make
such a conclusion? - ✔✔Using the notation of the notes, we want to make sure to get
the right answer with probability of P ⋆ = 0.95 whenever μ [ k ] − μ [ k − 1 ] ≥ δ ⋆ = 1.
We simply go to NB's table with k = 3 and δ ⋆ / σ = to obtain a sample size of n =
30 from each population.
✔✔In the above problem, suppose that we take the necessary observations and we
come up with the following sample means: X ¯ 1 = 7.6, X ¯ 2 = 11.1, and X ¯ 3 = 3.6.
What do we do? - ✔✔Pick population 2 and say that we are right with probability at least
95%
✔✔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? - ✔✔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? - ✔✔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? - ✔✔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. - ✔✔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. - ✔✔[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? - ✔✔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. - ✔✔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
