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Practice Final Exams ISYE6644 WITH ACTUAL CORRECT DETAILED ANSWERS

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Practice Final Exams ISYE6644 WITH ACTUAL CORRECT DETAILED ANSWERSPractice Final Exams ISYE6644 WITH ACTUAL CORRECT DETAILED ANSWERSPractice Final Exams ISYE6644 WITH ACTUAL CORRECT DETAILED ANSWERSPractice Final Exams ISYE6644 WITH ACTUAL CORRECT DETAILED ANSWERSPractice Final Exams ISYE...

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  • October 10, 2024
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  • ISYE6644
  • ISYE6644
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DrJudy
CURRENTLY TESTING SOLUTIONS Practice Final Exams
ISYE6644 WITH ACTUAL CORRECT DETAILED ANSWERS




TRUE or FALSE? Suppose that X1, X2,... is a stationary stochastic process with covariance function Rk =
Cov(X1, X1+k), for k=0,1,... Then the variance of the sample mean can be represented as Var(X) = 1/n[Ro
+ 2(1-k/n)Rk] - ANSWER-TRUE



TRUE or FALSE? 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.) 2



Show how to generate in Arena a discrete random variable X for which we have Pr(X = x) = 0.3 if x = −3
0.6 if x = 3.5 0.1 if x = 4 0 otherwise. - ANSWER-DISC(0.3, −3, 0.9, 3.5, 1.0, 4)



TRUE or FALSE? 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.



TRUE or FALSE? An entity can be scheduled to visit the same resource twice, with different service time
distributions on the two visits! - ANSWER-TRUE



TRUE or FALSE? Arena has a built-in Input Analyzer tool that allows for the fitting of certain distributions
to data. - ANSWER-TRUE



Suppose the continuous random variable X has p.d.f. f(x) = 2x for 0 ≤ x ≤ 1. Find the inverse of X's c.d.f.,
and thus show how to generate the RV X in terms of a Unif(0,1) PRN U. - ANSWER-X=sqrt(U)

The c.d.f. is easily shown to be F(x) = x 2 for 0 ≤ x ≤ 1, so that the Inverse Transform Theorem gives F(X) =
X2 = U ∼ Unif(0, 1). Solving for X, we obtain the desired inverse, F −1 (U) = X = √ U, where we don't worry
about the negative square root, since X ≥ 0. Thus, (d) is the answer.

,If U1 and U2 are i.i.d. Unif(0,1) with U1 = 0.45 and U2 = 0.45, use Box-Muller to generate two i.i.d.
Nor(0,1) realizations. - ANSWER-Z1 = -1.2019, Z2 = 0.3905



Suppose that Z1, Z2, and Z3 are i.i.d. Nor(0,1) random variables, and let T = Z1 /sqrt((Z 2 2 + Z 2 3 )/2) .
Find the value of x such that Pr(T < x) = 0.99. - ANSWER-x=6.965



Suppose X has the Laplace distribution with p.d.f. f(x) = λ/2 e^−λ|x| for x ∈ R and λ > 0. This looks like
two exponentials symmetric on both sides of the yaxis. Which of the methods below would be very
reasonable to use to generate realizations from this distribution? - ANSWER-Inverse Transform Method
AND Acceptance-Rejection



Consider a bivariate normal random variable (X, Y ), for which E[X] = −3, Var(X) = 4, E[Y ] = −2, Var(Y ) = 9,
and Cov(X, Y ) = 2. Find the Cholesky matrix associated with (X, Y ), i.e., the lower-triangular matrix C
such that Σ = CC0 , where Σ is the variance-covariance matrix. - ANSWER-C = (2 0

1 2sqrt(2))



Consider a nonhomogeneous Poisson arrival process with rate function λ(t) = 2t for t ≥ 0. Find the
probability that there will be exactly 2 arrivals between times t = 1 and 2. - ANSWER-0.224



Suppose we are generating arrivals from a nonhomogeneous Poisson process with rate function λ(t) = 1
+ sin(πt), so that the maximum rate is λ ? = 2, which is periodically achieved. Suppose that we generate a
potential arrival (i.e., one at rate λ ? ) at time t = 0.75. What is the probability that our usual thinning
algorithm will actually accept that potential arrival as an actual arrival? (Note that the π means that
calculations are in radians.) - ANSWER-0.854



Suppose X1, X2, . . . is an i.i.d. sequence of random variables with mean µ and variance σ 2 . Consider the
process Yn(t) ≡ Pbntc i=1 (Xi − µ)/(σ √ n) for t ≥ 0. What is the asymptotic probability that Yn(4) will be at
least 2 as n becomes large? Hint: Recall that Donsker's Theorem states that Yn(t) converges to a standard
Brownian motion as n becomes large. - ANSWER-0.1587



Which one of the following properties of a Brownian motion process W(t) is FALSE? - ANSWER-W(3) −
W(1) is independent of W(4) − W(2).



Find the sample variance of −10, 10, 0. - ANSWER-100

, S^2 = 100



If X1, . . . , X10 are i.i.d. Exp(1/7) (i.e., having mean 7), what is the expected value of the sample variance
S 2 ? - ANSWER-49

S^2 is always unbiased for the variance of Xi. Thus, we have E[S^2] = Var(Xi) = 1/lambda^2 = 49.



TRUE or FALSE? The mean squared error of an estimator is the square of the bias plus the square of its
variance - ANSWER-False



If X1 = 7, X2 = 3, and X3 = 5 are i.i.d. realizations from a Nor(µ, σ2 ) distribution, what is the value of the
maximum likelihood estimate for the variance σ 2 ? - ANSWER-2.667



Suppose that we take three i.i.d. observations X1 = 2, X2 = 3, and X3 = 1 from X ∼ Exp(λ). Using the
maximum likelihood estimate for λ, find the MLE of Pr(X > 2). - ANSWER-0.368



Suppose we're conducting a χ 2 goodness-of-fit test to determine whether or not 100 i.i.d. observations
are from a Johnson distribution with s = 4 unknown parameters a, b, c, and d. (The Johnson distribution
is very general and often fits data quite well.) If we divide the observations into k = 10 equal-probability
intervals and we observe a g-o-f statistic of χ 2 0 = 14.2, will we ACCEPT (i.e., fail to reject) or REJECT the
null hypothesis of the Johnson? Use level of significance α = 0.05 for your test. - ANSWER-Reject. Not
that the x^2 test has v = k-s-1 = 10-4-1 = 5 degrees of freedom. Then x0^2 = 14.2 > x0.05,5^2 = 11.07.



TRUE or FALSE? 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



Let's run a simulation whose output is a sequence of consecutive customer waiting times in a crowded
store. Which of the following statements is true? - ANSWER-The waiting times are correlated.



Suppose we want to estimate the expected average waiting time (in minutes) for the first m = 100
customers at a bank. We make r = 3 independent replications of the system, each initialized empty and
idle and consisting of 100 waiting times. The resulting replicate means are: 12, 14, 11. Find a 95% two-
sided confidence interval for the mean average waiting time for the first 100 customers. - ANSWER-[8.5,
16.1]

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