ISYE-6644 Simulation/ Latest Update
What are i.i.d. random variables? - CORRECT ANSWERS -It means "Independent and
identically distributed".
A good example is a succession of throws of a fair coin: The coin has no memory, so all
the throws are "independent".
And every throw is 50:50 (heads: tails), so the coin is and stays fair - the distribution
from which every throw is drawn, so to speak, is and stays the same: "identically
distributed".
(9.2) TRUE or FALSE? Suppose that X1,X2,...,Xn are consecutive waiting times, and
we define the sample mean X¯=∑Xi/n. Then Var(X¯)=Var(Xi)/n. - CORRECT
ANSWERS -False. 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!)
(9.4) TRUE or FALSE? You can also conduct finite-horizon estimation for quantities
other than expected values, e.g., simulate a bank from 8:00 a.m. to 5:00 p.m., and find
a confidence interval for the 95th quantile of customer waiting times. - CORRECT
ANSWERS -True
(9.5) How can we deal with initialization bias if we want to do a steady-state analysis? -
CORRECT ANSWERS -Make an extremely long run in order to overwhelm it. Also,
Truncate (delete) some of the initial data.
(9.6) Which scenarios might be well-suited for a steady-state analysis? - CORRECT
ANSWERS -1) Simulate an assembly line working 24/7. 2) A Markov chain simulated
until the transition probabilities appear to converge.
(9.6) The method of batch means - CORRECT ANSWERS -The resulting batch sample
means are approximately i.i.d. normal.
(9.7) True or False. The method of batch means is easy to use. - CORRECT
ANSWERS -True
(9.7) True or False. Batch means chops the consecutive observations into a number of
nonoverlapping, contiguous batches. - CORRECT ANSWERS -True
(9.7) True or False. You can use the method of batch means to obtain a confidence
interval for the steady-state mean μ. - CORRECT ANSWERS -True
(9.7) True or False. The batch means estimator for the variance parameter σ^2 is
asymptotically unbiased as the batch size m→∞. - CORRECT ANSWERS -True
, (10.1) Which of the following parameters can you get confidence intervals for? Means,
Variances, Quantiles, Differences between the means of two systems, or all of those. -
CORRECT ANSWERS -All. We can get CIs for means, variances, quantiles, and
differences between the means of two systems.
Bernoulli probability selection problem - CORRECT ANSWERS -Bunch of Bernoulli
populations and find the one with the best success probability
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - CORRECT ANSWERS -14/3 (or
4.666). If sample is entire population than variance is 4.
(8.1) M/M/1 queue - CORRECT ANSWERS -queue length having a single server.
(8.3) If the expected value of your estimator equals the parameter that you're trying to
estimate, then your estimator is unbiased. True of False - CORRECT ANSWERS -True.
This is the definition of unbiasedness
(8.3) If X1, X2, ..., Xn are i.i.d. with mean mu, then the sample mean X-bar is unbiased
for mu. True or False - CORRECT ANSWERS -True.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - CORRECT ANSWERS
-Bias^2 + Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? -
CORRECT ANSWERS -λ is the mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? -
CORRECT ANSWERS -λ is the sample variance and the mean
(8.4) 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?
- CORRECT ANSWERS -B is lower. Bias^2 + Variance: 18 < 21
MLE - CORRECT ANSWERS -Maximum Likelihood Estimator - "A method of estimating
the parameters of a distribution by maximizing a likelihood function, so that under the
assumed statistical model the observed data is most probable."
(8.4) Suppose that X1=4, X2=3, X3=5 are i.i.d. realizations from an Exp(λ) distribution.
What is the MLE of λ? - CORRECT ANSWERS -0.25
(8.5/8.6) If X1=2, X2=−2, and X3=0 are i.i.d. realizations from a Nor(μ , σ^2) distribution,
what is the value of the maximum likelihood estimate for the variance σ^2? - CORRECT
ANSWERS -8/3. MLE of σ^2 is the summation of the squared differences (Xi - μ), all
divided by n.
What are i.i.d. random variables? - CORRECT ANSWERS -It means "Independent and
identically distributed".
A good example is a succession of throws of a fair coin: The coin has no memory, so all
the throws are "independent".
And every throw is 50:50 (heads: tails), so the coin is and stays fair - the distribution
from which every throw is drawn, so to speak, is and stays the same: "identically
distributed".
(9.2) TRUE or FALSE? Suppose that X1,X2,...,Xn are consecutive waiting times, and
we define the sample mean X¯=∑Xi/n. Then Var(X¯)=Var(Xi)/n. - CORRECT
ANSWERS -False. 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!)
(9.4) TRUE or FALSE? You can also conduct finite-horizon estimation for quantities
other than expected values, e.g., simulate a bank from 8:00 a.m. to 5:00 p.m., and find
a confidence interval for the 95th quantile of customer waiting times. - CORRECT
ANSWERS -True
(9.5) How can we deal with initialization bias if we want to do a steady-state analysis? -
CORRECT ANSWERS -Make an extremely long run in order to overwhelm it. Also,
Truncate (delete) some of the initial data.
(9.6) Which scenarios might be well-suited for a steady-state analysis? - CORRECT
ANSWERS -1) Simulate an assembly line working 24/7. 2) A Markov chain simulated
until the transition probabilities appear to converge.
(9.6) The method of batch means - CORRECT ANSWERS -The resulting batch sample
means are approximately i.i.d. normal.
(9.7) True or False. The method of batch means is easy to use. - CORRECT
ANSWERS -True
(9.7) True or False. Batch means chops the consecutive observations into a number of
nonoverlapping, contiguous batches. - CORRECT ANSWERS -True
(9.7) True or False. You can use the method of batch means to obtain a confidence
interval for the steady-state mean μ. - CORRECT ANSWERS -True
(9.7) True or False. The batch means estimator for the variance parameter σ^2 is
asymptotically unbiased as the batch size m→∞. - CORRECT ANSWERS -True
, (10.1) Which of the following parameters can you get confidence intervals for? Means,
Variances, Quantiles, Differences between the means of two systems, or all of those. -
CORRECT ANSWERS -All. We can get CIs for means, variances, quantiles, and
differences between the means of two systems.
Bernoulli probability selection problem - CORRECT ANSWERS -Bunch of Bernoulli
populations and find the one with the best success probability
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - CORRECT ANSWERS -14/3 (or
4.666). If sample is entire population than variance is 4.
(8.1) M/M/1 queue - CORRECT ANSWERS -queue length having a single server.
(8.3) If the expected value of your estimator equals the parameter that you're trying to
estimate, then your estimator is unbiased. True of False - CORRECT ANSWERS -True.
This is the definition of unbiasedness
(8.3) If X1, X2, ..., Xn are i.i.d. with mean mu, then the sample mean X-bar is unbiased
for mu. True or False - CORRECT ANSWERS -True.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - CORRECT ANSWERS
-Bias^2 + Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? -
CORRECT ANSWERS -λ is the mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? -
CORRECT ANSWERS -λ is the sample variance and the mean
(8.4) 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?
- CORRECT ANSWERS -B is lower. Bias^2 + Variance: 18 < 21
MLE - CORRECT ANSWERS -Maximum Likelihood Estimator - "A method of estimating
the parameters of a distribution by maximizing a likelihood function, so that under the
assumed statistical model the observed data is most probable."
(8.4) Suppose that X1=4, X2=3, X3=5 are i.i.d. realizations from an Exp(λ) distribution.
What is the MLE of λ? - CORRECT ANSWERS -0.25
(8.5/8.6) If X1=2, X2=−2, and X3=0 are i.i.d. realizations from a Nor(μ , σ^2) distribution,
what is the value of the maximum likelihood estimate for the variance σ^2? - CORRECT
ANSWERS -8/3. MLE of σ^2 is the summation of the squared differences (Xi - μ), all
divided by n.
