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ISYE 6644 SIMULATION MIDTERM EXAM
NEWEST 2025/2026 COMPLETE QUESTIONS AND
CORRECT ANSWERS (VERIFIED ANSWERS)
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If f(x, y) = cxy for all 0 < x < 1 and 1 < y < 2, where c is whatever value makes this thing integrate to 1, then X and Y are independent random variables ......answer.....TRUE. (Because f(x, y) = a(x)b(y) factors nicely, and there are no funny limits.) In our Arena Call Center example, it was possible for entities to be left in the system when it shut down at 7:00 p.m. (even though we stopped allowing customers to enter the system at 6:00 p.m.). ......answer.....TRUE — because of the small chance that a callback will occur. An entity can be scheduled to visit the same resource twice, with different service time distributions on the two visits! ......answer.....TRUE Arena has a built-in Input Analyzer tool that allows for the fitting of certain distributions to data. ......answer.....TRUE
The Kolmogorov-Smirnov test can be used both to see (i) if data seem to fit to a particular hypothesized distribution and (ii) if the data are independent. ......answer.....FALSE. (It's just a goodness-of-fit test.) Welch's method is a graphical technique to estimate truncation (initialization bias) points for steady-state simulation. ......answer.....TRUE If you are using a ranking-and-selection procedure and two competitors happen to fall within the indifference-zone, then you don't really care too much which one you end up selecting. ......answer.....TRUE. That's why it's called the IZ! TRUE or FALSE? Sequential ranking-and-selection procedures are designed to stop early if one alternative seems to be way out in front of the others. ......answer.....TRUE Suppose, when designing a ranking-and-selection procedure, you have decided to increase the desired probability of correct selection compared to a previous run of the procedure. What can you expect? (a) Sample sizes that are about the same (b) Larger sample sizes
TRUE or FALSE? In Arena, the expression NORM(-1,2)+NORM(3,2) has the same distribution as the expression NORM(2,4) ......answer.....FALSE. The above expression evaluates to NORM(2,√ 8) since, in Arena, NORM(x,2) corresponds to a normal random variate with a mean of x and a standard deviation of 2; so the variance of this random variable is 22 = 4. TRUE or FALSE? In Arena, it is only possible to use one resource at a time in a PROCESS module. ......answer.....FALSE TRUE or FALSE? In Arena, once an entity is defined, it is not possible to change the priority in which it gets processed by a resource. ......answer.....FALSE TRUE or FALSE? In Arena, it is not possible to use a single RELEASE module to release multiple different resources ......answer.....FALSE. We observed this in the ReleaseHospitalRoomLater demo in HW What is the distribution given by the Arena expression UNIF(1,2)+UNIF(1,2)? ......answer.....Tria(2,3,4).
An Arena ASSIGN module can be used to assign values to attributes and variables ......answer.....TRUE TRUE or FALSE? In Arena, there are several ways to terminate a simulation, for example, when a given simulation run length has been reached, or when a given number of entities have been created, or when a specific stopping criterion as be met. ......answer.....TRUE Suppose X is a discrete random variable that can take on four possible values: −0.1, 0, 2, and 4. The associated probabilities are, respectively: 0.3, 0.1, 0.2, and 0.4. Find an expression in Arena to generate one observation coming from this p.m.f. ......answer.....DISC(0.3, −0.1, 0.4, 0, 0.6, 2, 1, 4) DISC(F(x1), x1, F(x2), x2,.. .) Suppose there are 3 people in the line called joe.queue and 5 people in the line called tom.queue. What is the value of the following Arena expression? (NQ(joe.queue) > NQ(tom.queue)) + (TNOW >= 0 ......answer.....The logical expression (NQ(joe.queue) > NQ(tom.queue)) is false (because 3 ≤ 5) and therefore 0. The logical expression (TNOW >= 0) is always true (since time is always at least 0) and therefore 1. Thus, 0 + 1 = 1
TRUE or FALSE? In Arena, an alternative expression to generate exponential interarrival times having a mean of 3 is - (1/3)*LN(UNIF(0,1)) ......answer.....FALSE. This is the Inverse Transform Method to generate an observation coming from Exp(λ = 3), which has a mean of 1/3. TRUE or FALSE? In Arena, it is possible to set the maximum capacity of a queue from within a QUEUE block found in the Blocks template. ......answer.....TRUE. In fact, it's pretty much the only way. What is the variance of the random variable given in the Arena expression NORM(-1,3)+NORM(3,4)? ......answer.....: In Arena, NORM(x, y) corresponds to a normal random variate with a mean of x and a standard deviation of y. So the variance in question is 3 2 + 42 = 25. TRUE or FALSE? In Arena, it is possible to grab more than one resource at a time in a PROCESS module ......answer.....TRUE TRUE or FALSE? In Arena, it is possible to change the priority in which entities get processed by a resource ......answer.....TRUE
TRUE or FALSE? In Arena, it is possible to use a single RELEASE module to release multiple different resources ......answer.....TRUE. We observed this in the ReleaseHospitalRoomLater demo in HW What is the distribution given by the Arena expression UNIF(0,2)+UNIF(0,2)? ......answer.....Tria(0,2,4) TRUE or FALSE? An Arena ASSIGN module can be used to assign values to attributes but not variables. ......answer.....FALSE. It can be used for both Suppose X is a discrete random variable that can take on four possible values: −2, 1, 4, and 7. The associated probabilities are, respectively: 0.1, 0.3, 0.2, and 0.4. Find an expression in Arena to generate one observation coming from this p.m.f. ......answer.....DISC(−2, 0.1, 1, 0.4, 4, 0.6, 7, 1) Suppose there are 3 people in the line called joe.queue and 2 people in the line called tom.queue. What is the value of the following Arena expression? NQ(tom.queue) NQ(joe.queue) + (TNOW < −1) ......answer.....The logical expression (TNOW < - 1) is false (since time is always at least 0). Thus, NQ(tom.queue) NQ(joe.queue) − (TNOW < −1) = 6 + 0 = 6.
TRUE or FALSE? Suppose that U1, U2,... are truly i.i.d. U(0,1) random variables. Then a 95% chi-square goodness-of-fit test for uniformity will incorrectly reject uniformity of the observations about 5% of the time. ......answer.....TRUE. (That's what Type I error is.) TRUE or FALSE? Suppose Z1,... , Z6 are i.i.d. standard normal random variables obtained by the Box-Muller method. Then P6 i=1 Z 2 i Erlang3(1/2). ......answer.....TRUE. The sum of 6 i.i.d. Z 2 i random variables is a χ 2 (6), which is itself an Erlang3(1/2).
11 | TRUE or FALSE? If we want to generate X Pois(4.5), then it's better to use a normal approximation than acceptance-rejection. ......answer.....FALSE. (The normal approximation requires larger λ for the asymptotics to work and the efficiency to kick in.) TRUE or FALSE? The acceptance-rejection's majorizing function t(x) is usually a p.d.f. ......answer.....FALSE. (It integrates to something > 1.) If X1,... , Xn are i.i.d. Exp(1/9), what is the expected value of the sample variance S 2? ......answer.....E[S^2 ] = σ^2 = 81. Find x such that e 2x = 1/x. (Get within two decimals) ......answer.....Bisection search quickly reveals that x is about 0. TRUE or FALSE? The square root of the sample variance is unbiased for the standard deviation ......answer.....FALSE. (E[S 2 ] = σ 2 ; E[S] = σ.) Which is usually a better way to deal with initialization bias in steady-state simulation analysis: (i) make an extremely long run to overwhelm the bias, or (ii) perform truncation? ......answer.....Truncate
13 | 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. ......answer.....True 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. ......answer.....True For which scenarios(s) below might it be appropriate to use a Bernoulli selection procedure? a) Find the inventory policy having the largest profit. b) Find the drug giving the best chance of a cure. c) Find the maintenance policy having the lowest failure probability. d) Find the scheduling rule that that has the best chance of making an ontime delivery. ......answer.....All three of (b), (c), and (d). Suppose that a Bernoulli selection procedure tells you to take 100 observations from each of two populations, A and B. It turns out that A gets
85 successes and B gets 46 successes. What do you think? ......answer.....1) A almost certainly has a higher success probability than B.
- We could've probably stopped sampling a bit earlier (i.e., with fewer than 100 observations) because A was so far ahead of B.
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. ......answer.....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.} 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. ......answer.....TRUE TRUE or FALSE? The common random numbers technique intentionally induces positive correlation between two systems - much like a paired- confidence interval. ......answer.....TRUE CRN depends on someone's ability to manipulate the underlying pseudorandom numbers - e.g., use the same arrival times when simulating two competing simulated systems. So who ultimately controls those PRNs?? ......answer.....You do - you are powerful!
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. ......answer.....TRUE TRUE or FALSE? The control variates technique provides unbiased, lowvariance estimates using a method reminiscent of regression. ......answer.....TRUE Statistical ranking and selection techniques have been designed to address a variety of comparison problems. Which ones from the following list? Find the population having the largest mean. Find the system with the smallest variance. Find the alternative with the highest success probability. Find the most-popular candidate. All of the above. ......answer.....All of the above. Suppose we are dealing with i.i.d. normal observations with unknown variance. Which of the following is true about a 95% confidence interval for the mean μ? ......answer.....We are 95% sure that our CI will actually contain the unknown value of μ.
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? ......answer.....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? ......answer.....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 = 1 / 2 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? ......answer.....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-secondhighest 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? ......answer.....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 finitehorizon simulation? ......answer.....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? ......answer.....The consecutive daily inventory levels may not be identically distributed.
