ISYE-6644 Simulation

(8.3) Find the sample variance of -3, -2, -1, 0, 1, 2, 3

14/3 (or 4.666). If sample is entire population than variance is 4.

(8.1) M/M/1 queue

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

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

True.

(8.4) What is the MSE (Mean Squared Error) of an estimator?

Bias^2 + Variance

(8.3) What is the expected value of the mean of a Pois(λ) random variable?

λ is the mean and the variance

(8.3) What is the expected sample variance s^2 of a Pois(λ) random variable?

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?

B is lower. Bias^2 + Variance: 18 < 21

MLE

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?

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?

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?

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.

(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)?

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.

True

(8.6) TRUE or FALSE? Sometimes it might be difficult to obtain an MLE in closed form.

True. (There is a gamma example.)

(8.7) Suppose that the MLE for a parameter θ is θhat=4. Find the MLE for √θ.

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]?

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?

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?

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?

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?

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!

True

(9.1) We often distinguish between two general types of simulations with regard to output analysis. What are they called?

Finite-horizon and steady-state

What are i.i.d. random variables?

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.

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.

True

(9.5) How can we deal with initialization bias if we want to do a steady-state analysis?

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?

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

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.

True

(9.7) True or False. Batch means chops the consecutive observations into a number of nonoverlapping, contiguous batches.

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.

True

(9.7) True or False. The batch means estimator for the variance parameter σ^2 is asymptotically unbiased as the batch size m→∞.

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.

All. We can get CIs for means, variances, quantiles, and differences between the means of two systems.

Bernoulli probability selection problem

Bunch of Bernoulli populations and find the one with the best success probability

Multinomial cell selection problem

Normal means ranking and selection problem

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.

True.

(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?

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.

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.`

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.

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 two competing simulated systems. So who ultimately controls those PRNs??

You do

Anithetic Random Numbers Method

Used for negative correlation between estimators. Opposite of CRN

CRN

Common Random Numbers

(10.7) TRUE or FALSE? The antithetic random numbers technique intentionally induces negative correlation between two runs of the same system - this allows us to better estimate the mean of the system.

True

(10.8) TRUE or FALSE? The control variates technique provides unbiased, low-variance estimates using a method reminiscent of regression.

True

(10.9) Statistical ranking and selection techniques have been designed to address a variety of comparison problems. Which ones from the following list? (a) Find the population having the largest mean.

(b) Find the system with the smallest variance.

(c) Find the alternative with the highest success probability.

(d) Find the most-popular candidate.

All of them

(10.10) What is a possible goal of an indifference-zone normal means selection technique?

Find the normal population having the largest mean, especially if the largest mean is ≫ the second-largest. Using the notation of the notes, we want to make sure to get the right answer in the case that μk − μk−1 ≥ δ⋆.

(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.

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.

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?

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 24/7/365

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

c. Simulating the operations of a bank from 9:00 a.m. until 5:00 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.

d. The consecutive daily inventory levels may not be identically distributed.

(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

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

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

d. Negative Binomial

(Lesson 8.2: Identifying Distributions.) Name That Distribution!

IQs

a. Uniform

b. Normal

c. Exponential

d. Weibull

e. Pareto

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

c. Triangular