1
PRACTICE FINAL ISYE 6644 EXAM LATEST VERSION -2025/2026-
100+ QUESTIONS AND VERIFIED ANSWERS ALL THE BEST
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.
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.).
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!
TRUE
TRUE or FALSE? Arena has a built-in Input Analyzer tool that allows for the fitting
of certain distributions to data.
TRUE
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]
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.
TRUE. (Because f(x, y) = a(x)b(y) factors nicely, and there are no funny limits.) 2
, 2
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.
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.
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.
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?
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.
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.
, 3
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.)
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.
0.1587
Which one of the following properties of a Brownian motion process W(t) is
FALSE?
W(3) − W(1) is independent of W(4) − W(2).
Find the sample variance of −10, 10, 0.
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 ?
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
PRACTICE FINAL ISYE 6644 EXAM LATEST VERSION -2025/2026-
100+ QUESTIONS AND VERIFIED ANSWERS ALL THE BEST
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.
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.).
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!
TRUE
TRUE or FALSE? Arena has a built-in Input Analyzer tool that allows for the fitting
of certain distributions to data.
TRUE
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]
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.
TRUE. (Because f(x, y) = a(x)b(y) factors nicely, and there are no funny limits.) 2
, 2
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.
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.
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.
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?
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.
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.
, 3
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.)
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.
0.1587
Which one of the following properties of a Brownian motion process W(t) is
FALSE?
W(3) − W(1) is independent of W(4) − W(2).
Find the sample variance of −10, 10, 0.
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 ?
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
