ISYE 6644 FINAL EXAM 2024 NEWEST 2 VERSIONS
(VERSION A AND B) COMPLETE ACCURATE EXAM
QUESTIONS WITH DETAILED VERIFIED ANSWERS
(100% CORRECT ANSWERS) /ALREADY GRADED A+
VERSION A
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] - ANSWERTRUE
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. -
ANSWERTRUE. (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. - ANSWERDISC(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.). - ANSWERTrue - 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! - ANSWERTRUE
TRUE or FALSE? Arena has a built-in Input Analyzer tool that allows for the fitting of
certain distributions to data. - ANSWERTRUE
,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. - ANSWERX=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. - ANSWERZ1 = -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. - ANSWERx=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? - ANSWERInverse 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. - ANSWERC = (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. -
ANSWER0.224
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.) - ANSWER0.854
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. - ANSWER0.1587
Which one of the following properties of a Brownian motion process W(t) is FALSE? -
ANSWERW(3) − W(1) is independent of W(4) − W(2).
Find the sample variance of −10, 10, 0. - ANSWER100
, 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 ? - ANSWER49
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 - ANSWERFalse
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 ? - ANSWER2.667
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). -
ANSWER0.368
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. -
ANSWERReject. 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. -
ANSWERFalse
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? - ANSWERThe
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]
Suppose that µ ∈ [−30, 90] is a 90% confidence interval for the mean cost incurred by a
certain inventory policy. Further suppose that this interval was based on 4 independent
replications of the underlying inventory system. Unfortunately, the boss has decided that
she wants a 95% confidence interval. Can you supply it? - ANSWER[−51.14, 111.14].
(VERSION A AND B) COMPLETE ACCURATE EXAM
QUESTIONS WITH DETAILED VERIFIED ANSWERS
(100% CORRECT ANSWERS) /ALREADY GRADED A+
VERSION A
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] - ANSWERTRUE
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. -
ANSWERTRUE. (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. - ANSWERDISC(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.). - ANSWERTrue - 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! - ANSWERTRUE
TRUE or FALSE? Arena has a built-in Input Analyzer tool that allows for the fitting of
certain distributions to data. - ANSWERTRUE
,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. - ANSWERX=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. - ANSWERZ1 = -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. - ANSWERx=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? - ANSWERInverse 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. - ANSWERC = (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. -
ANSWER0.224
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.) - ANSWER0.854
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. - ANSWER0.1587
Which one of the following properties of a Brownian motion process W(t) is FALSE? -
ANSWERW(3) − W(1) is independent of W(4) − W(2).
Find the sample variance of −10, 10, 0. - ANSWER100
, 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 ? - ANSWER49
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 - ANSWERFalse
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 ? - ANSWER2.667
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). -
ANSWER0.368
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. -
ANSWERReject. 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. -
ANSWERFalse
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? - ANSWERThe
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]
Suppose that µ ∈ [−30, 90] is a 90% confidence interval for the mean cost incurred by a
certain inventory policy. Further suppose that this interval was based on 4 independent
replications of the underlying inventory system. Unfortunately, the boss has decided that
she wants a 95% confidence interval. Can you supply it? - ANSWER[−51.14, 111.14].
