ISYE-6644 COMPLETE LATEST
SIMULATION BASED ACTUAL SET WITH
QUESTIONS AND CORRECT/VERIFIED
SOLUTIONS
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - correct answer-✅14/3 (or 4.666). If sample is
entire population than variance is 4.
(8.1) M/M/1 queue - correct answer-✅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 answer-✅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 answer-✅True.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - correct answer-✅Bias^2 + Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? - correct answer-✅λ is the
mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? - correct answer-✅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 answer-✅B is lower. Bias^2
+ Variance: 18 < 21
MLE - correct answer-✅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 answer-✅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 answer-✅8/3. MLE of σ^2 is the
summation of the squared differences (Xi - μ), all divided by n.
(8.5/8.6) Suppose we observe the Pois(λ) realizations X1=5, X2=9 and X3=1. What is the maximum
likelihood estimate of λ? - correct answer-✅5. λ is estimated as the summation of sample values divided
by the number of sample values. (5+9+1)/3 = 5
(8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE for p. - correct answer-✅
(8.7) Suppose that we have a number of observations from a Pois(λ) distribution, and it turns out that
the MLE for λ is λhat=5. What's the maximum likelihood estimate of Pr(X=3)? - correct answer-✅0.1404.
P(X=x) = λ^x * e^(−λ) / x!
(8.6) TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g., for the Nor(μ,σ2)
distribution. - correct answer-✅True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. - correct answer-
✅True. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. - correct answer-✅2.
Invariance immediately implies that the MLE of √θ is simply √θhat = 2
(8.8) Suppose that we observe X1 = 5, X2 = 9, and X3 = 1. What's the method of moments estimate of
E[X^2]? - correct answer-✅35.6667. Second moment is the sum of the squared samples divided by the
number of samples. (5^2 + 9^2 + 1^2) / 3 = 35.666666667
, (8.9) Suppose we're conducting a χ^2 goodness-of-fit test with Type I error rate α = 0.01 to determine
whether or not 100 i.i.d. observations are from a lognormal distribution with unknown parameters μ and
σ^2. If we divide the observations into 5 equal-probability intervals and we observe a g-o-f statistic of
χ0^2 = 11.2, will we ACCEPT (i.e., fail to reject) or REJECT the null hypothesis of lognormality? - correct
answer-✅Reject. k = 5, subtract 1 and subtract 2 for the two unknown parameters (or had to estimate),
so degrees of freedom is 2. critical value for dof 2 and alpha 0.01 is 9.21. 11.2 is not smaller than 9.21 so
we reject it. Not a good fit.
(8.9) Suppose H0 is true, but you've just rejected it! What have you done? - correct answer-✅Type I
error
(8.10/8.11) The test statistic is χ0^2 = 9.12. Now, let's use our old friend α = 0.05 in our test. Let k = 4
denote the number of cells (that we ultimately ended up with) and let s = 1 denote the number of
parameters we had to estimate. Then we compare against χ^2(α=0.05 , k − s − 1) = χ^2(α=0.05 , 2) =
5.99. Do we ACCEPT (i.e., fail to reject) or REJECT the Geometric hypothesis? - correct answer-✅Reject.
The test statistic 9.12 is not less than 5.99.
(8.12) Consider the PRN's U1 = 0.1 , U2 = 0.9 , and U3 = 0.2. Use Kolmogorov-Smirnov with α = 0.05 to
test to see if these numbers are indeed uniform. Do we ACCEPT (i.e., fail to reject) or REJECT uniformity?
- correct answer-✅Accept. From table, D(α=0.05, 3) = 0.70760. Create ordered sample set: 0.1, 0.2, 0.9.
Since the max value of D test is 0.467, then we fail to reject because it is smaller.
(9.1) 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
(9.1) We often distinguish between two general types of simulations with regard to output analysis.
What are they called? - correct answer-✅Finite-horizon and steady-state
What are i.i.d. random variables? - correct answer-✅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 answer-✅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 answer-✅True
(9.5) How can we deal with initialization bias if we want to do a steady-state analysis? - correct answer-
✅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 answer-✅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 answer-✅The resulting batch sample means are
aproximately i.i.d. normal.
(9.7) True or False. The method of batch means is easy to use. - correct answer-✅True
(9.7) True or False. Batch means chops the consecutive observations into a number of nonoverlapping,
contiguous batches. - correct answer-✅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 answer-✅True
SIMULATION BASED ACTUAL SET WITH
QUESTIONS AND CORRECT/VERIFIED
SOLUTIONS
(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3 - correct answer-✅14/3 (or 4.666). If sample is
entire population than variance is 4.
(8.1) M/M/1 queue - correct answer-✅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 answer-✅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 answer-✅True.
(8.4) What is the MSE (Mean Squared Error) of an estimator? - correct answer-✅Bias^2 + Variance
(8.3) What is the expected value of the mean of a Pois(λ) random variable? - correct answer-✅λ is the
mean and the variance
(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable? - correct answer-✅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 answer-✅B is lower. Bias^2
+ Variance: 18 < 21
MLE - correct answer-✅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 answer-✅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 answer-✅8/3. MLE of σ^2 is the
summation of the squared differences (Xi - μ), all divided by n.
(8.5/8.6) Suppose we observe the Pois(λ) realizations X1=5, X2=9 and X3=1. What is the maximum
likelihood estimate of λ? - correct answer-✅5. λ is estimated as the summation of sample values divided
by the number of sample values. (5+9+1)/3 = 5
(8.5) Suppose X1, ..., Xn are i.i.d. Bern(p). Find the MLE for p. - correct answer-✅
(8.7) Suppose that we have a number of observations from a Pois(λ) distribution, and it turns out that
the MLE for λ is λhat=5. What's the maximum likelihood estimate of Pr(X=3)? - correct answer-✅0.1404.
P(X=x) = λ^x * e^(−λ) / x!
(8.6) TRUE or FALSE? It's possible to estimate two MLEs simultaneously, e.g., for the Nor(μ,σ2)
distribution. - correct answer-✅True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form. - correct answer-
✅True. (There is a gamma example.)
(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ. - correct answer-✅2.
Invariance immediately implies that the MLE of √θ is simply √θhat = 2
(8.8) Suppose that we observe X1 = 5, X2 = 9, and X3 = 1. What's the method of moments estimate of
E[X^2]? - correct answer-✅35.6667. Second moment is the sum of the squared samples divided by the
number of samples. (5^2 + 9^2 + 1^2) / 3 = 35.666666667
, (8.9) Suppose we're conducting a χ^2 goodness-of-fit test with Type I error rate α = 0.01 to determine
whether or not 100 i.i.d. observations are from a lognormal distribution with unknown parameters μ and
σ^2. If we divide the observations into 5 equal-probability intervals and we observe a g-o-f statistic of
χ0^2 = 11.2, will we ACCEPT (i.e., fail to reject) or REJECT the null hypothesis of lognormality? - correct
answer-✅Reject. k = 5, subtract 1 and subtract 2 for the two unknown parameters (or had to estimate),
so degrees of freedom is 2. critical value for dof 2 and alpha 0.01 is 9.21. 11.2 is not smaller than 9.21 so
we reject it. Not a good fit.
(8.9) Suppose H0 is true, but you've just rejected it! What have you done? - correct answer-✅Type I
error
(8.10/8.11) The test statistic is χ0^2 = 9.12. Now, let's use our old friend α = 0.05 in our test. Let k = 4
denote the number of cells (that we ultimately ended up with) and let s = 1 denote the number of
parameters we had to estimate. Then we compare against χ^2(α=0.05 , k − s − 1) = χ^2(α=0.05 , 2) =
5.99. Do we ACCEPT (i.e., fail to reject) or REJECT the Geometric hypothesis? - correct answer-✅Reject.
The test statistic 9.12 is not less than 5.99.
(8.12) Consider the PRN's U1 = 0.1 , U2 = 0.9 , and U3 = 0.2. Use Kolmogorov-Smirnov with α = 0.05 to
test to see if these numbers are indeed uniform. Do we ACCEPT (i.e., fail to reject) or REJECT uniformity?
- correct answer-✅Accept. From table, D(α=0.05, 3) = 0.70760. Create ordered sample set: 0.1, 0.2, 0.9.
Since the max value of D test is 0.467, then we fail to reject because it is smaller.
(9.1) 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
(9.1) We often distinguish between two general types of simulations with regard to output analysis.
What are they called? - correct answer-✅Finite-horizon and steady-state
What are i.i.d. random variables? - correct answer-✅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 answer-✅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 answer-✅True
(9.5) How can we deal with initialization bias if we want to do a steady-state analysis? - correct answer-
✅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 answer-✅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 answer-✅The resulting batch sample means are
aproximately i.i.d. normal.
(9.7) True or False. The method of batch means is easy to use. - correct answer-✅True
(9.7) True or False. Batch means chops the consecutive observations into a number of nonoverlapping,
contiguous batches. - correct answer-✅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 answer-✅True
