Practice Final Exams ISYE6644 [ACTUAL EXAM] LATEST
VERSION [QUESTIONS AND ANSWERS] WITH PRACTICE
EXAM DETAILED AND VERIFIED FOR GUARANTEED PASS-
LATEST UPDATE 2025 GRADED A
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. - CORRECT 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.). - CORRECT 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! - CORRECT ANSWER TRUE
TRUE or FALSE? Arena has a built-in Input Analyzer tool that allows for the fitting of
certain distributions to data. - CORRECT 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. - CORRECT 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. - CORRECT 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. - CORRECT 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? - CORRECT 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. - CORRECT 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. -
CORRECT 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.) - CORRECT 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. - CORRECT ANSWER 0.1587
Which one of the following properties of a Brownian motion process W(t) is FALSE? -
CORRECT ANSWER W(3) − W(1) is independent of W(4) − W(2).
Find the sample variance of −10, 10, 0. - CORRECT 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 ? - CORRECT 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 - CORRECT 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 ? - CORRECT
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). -
CORRECT 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. - CORRECT
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. -
CORRECT 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? - CORRECT
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. - CORRECT 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? - CORRECT ANSWER
[−51.14, 111.14].
TRUE or FALSE? Welch's method is a graphical technique to estimate truncation
(initialization bias) points for steady-state simulation. - CORRECT ANSWER True
Suppose that we're studying a stochastic process whose covariance function is Rk =
3−k for k = 0, ±1, ±2, and 0 otherwise. Find the variance of X¯ 3 (the sample mean of
the first 3 observations). - CORRECT ANSWER 2.11
Consider the following 5 observations: 54 80 75 62 90 If we choose a batch size of 3,
calculate all of the overlapping batch means for me. - CORRECT ANSWER 69.7,
72.3, 75.7
Which variance reduction method uses the difference X¯ − Y¯ of two positively
correlated sample means to get a lower-variance estimator for the difference µX − µY of
the underlying unknown means? - CORRECT ANSWER Common random numbers.
VERSION [QUESTIONS AND ANSWERS] WITH PRACTICE
EXAM DETAILED AND VERIFIED FOR GUARANTEED PASS-
LATEST UPDATE 2025 GRADED A
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. - CORRECT 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.). - CORRECT 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! - CORRECT ANSWER TRUE
TRUE or FALSE? Arena has a built-in Input Analyzer tool that allows for the fitting of
certain distributions to data. - CORRECT 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. - CORRECT 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. - CORRECT 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. - CORRECT 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? - CORRECT 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. - CORRECT 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. -
CORRECT 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.) - CORRECT 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. - CORRECT ANSWER 0.1587
Which one of the following properties of a Brownian motion process W(t) is FALSE? -
CORRECT ANSWER W(3) − W(1) is independent of W(4) − W(2).
Find the sample variance of −10, 10, 0. - CORRECT 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 ? - CORRECT 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 - CORRECT 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 ? - CORRECT
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). -
CORRECT 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. - CORRECT
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. -
CORRECT 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? - CORRECT
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. - CORRECT 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? - CORRECT ANSWER
[−51.14, 111.14].
TRUE or FALSE? Welch's method is a graphical technique to estimate truncation
(initialization bias) points for steady-state simulation. - CORRECT ANSWER True
Suppose that we're studying a stochastic process whose covariance function is Rk =
3−k for k = 0, ±1, ±2, and 0 otherwise. Find the variance of X¯ 3 (the sample mean of
the first 3 observations). - CORRECT ANSWER 2.11
Consider the following 5 observations: 54 80 75 62 90 If we choose a batch size of 3,
calculate all of the overlapping batch means for me. - CORRECT ANSWER 69.7,
72.3, 75.7
Which variance reduction method uses the difference X¯ − Y¯ of two positively
correlated sample means to get a lower-variance estimator for the difference µX − µY of
the underlying unknown means? - CORRECT ANSWER Common random numbers.
