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ISYE 6644 – Simulation Examination
(Enhanced Version with High-Pass
Assurance).
(8.1) M/M/1 queue - Correct Answers>>>queue length having a single server.
(Lesson 8.2: Identifying Distributions.) Let's play Name That Distribution!
The number of times a "3" comes up in 10 dice tosses.
a. Bernoulli
b. Binomial
c. Geometric
d. Negative Binomial
e. Pareto - Correct Answers>>>b. Binomial
(Lesson 8.2: Identifying Distributions.) Name That Distribution!
The number of dice tosses until a 3 comes up.
a. Bernoulli
b. Binomial
c. Geometric
d. Negative Binomial
e. Pareto - Correct Answers>>>c. Geometric
(Lesson 8.2: Identifying Distributions.) Name That Distribution!
The number of dice tosses until a 3 comes up for the 4th time.
a. Bernoulli
b. Binomial
c. Geometric
d. Negative Binomial
e. Pareto - Correct Answers>>>d. Negative Binomial
(Lesson 8.2: Identifying Distributions.) Name That Distribution!
IQs
a. Uniform
b. Normal
c. Exponential
d. Weibull
e. Pareto - Correct Answers>>>b. Normal
(Lesson 8.2: Identifying Distributions.) Name That Distribution!
Cases in which you have limited information, e.g., you only know the min, max, and "most likely" values that a random variable can take.
a. Bernoulli
b. Poisson
c. Triangular
d. Weibull
e. Pareto - Correct Answers>>>c. Triangular
(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
(8.5/8.6) 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.
(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 Answers>>>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
Answers>>>
(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 Answers>>>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 Answers>>>True
(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in
closed form. - Correct Answers>>>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 Answers>>>2. Invariance immediately implies that the MLE of √θ is simply √θ hat = 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 Answers>>>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.
(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 Answers>>>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 Answers>>>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 Answers>>>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
Answers>>>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.
(9.6) 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 aproximately 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
Multinomial cell selection problem - Correct Answers>>>
Normal means ranking and selection problem - Correct Answers>>>Bunch of normal distributions and we want to find the one with the largest or smallest mean.
(10.2) "Assume unknown variance sigma^2". Probably will use t-distribution.
(10.2) If we have an i.i.d. normal sample of observations, X1,X2,...,Xn, what probability distribution is most-commonly used to obtain confidence
intervals for the mean? - Correct Answers>>>t-distribution
(10.4) 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. - Correct Answers>>>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.}
(10.5) TRUE or FALSE? You can use a version of independent replications to obtain confidence intervals for the difference in the means from two
simulation models.` - Correct Answers>>>TRUE. {It's pretty straightforward, though the notation is a little more tedious}.
(10.6) TRUE or FALSE? The common random numbers technique intentionally induces positive correlation between two systems - much like a paired-t
confidence interval. - Correct Answers>>>True
(10.6) CRN depends on someone's ability to manipulate the underlying pseudo-random numbers - e.g., use the same arrival times when simulating
(10.11) TRUE or FALSE? The Bechhofer procedure for selecting the normal population with the largest mean specifies the appropriate number of observations to take from each competing population, and simply selects the
competitor having the largest sample mean. - Correct Answers>>>True
(10.12) TRUE or FALSE? Sometimes a single-stage procedure like Bechhofer's is inefficient. In fact, it's possible to use certain sequential procedures that take observations one-at-a-time (instead of all at once in a single stage) to
make good selection decisions using fewer observations. - Correct
Answers>>>True
(10.16) 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. How many people does the single-stage procedure M{BEM} require us to
interview? - Correct Answers>>>From table, find P* = 0.9, Theta* = 1.4 and k = 3 competitors
(Lesson 9.1: Introduction to Output Analysis.) Which of the following problems might best be characterized by a finite-horizon simulation?
a. Simulating long-term hurricane patterns
b. Simulating a manufacturing cell 24/7/
c. Simulating the operations of a bank from 9:00 a.m. until 5:00 p.m.
d. Simulating the steady-state distribution of a Markov chain - Correct
Answers>>>c. Simulating the operations of a bank from 9:00 a.m. until 5: p.m.
(Lesson 9.1: Introduction to Output Analysis.) 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?
a. The consecutive daily inventory levels are independent.
b. The consecutive daily inventory levels are uncorrelated.
c. The consecutive daily inventory levels are normally distributed.
d. The consecutive daily inventory levels may not be identically distributed. -
Correct Answers>>>d. The consecutive daily inventory levels may not be identically distributed.
