ISYE6644 FINAL EXAM PRATICE TEST |2025-2026 LATEST UPDATED|
160 COMPREHENSIVE QUESTIONS AND ANSWERS | REAL AND
EXACT | 100% RATED CORRECT | 100% VERFIED | ALREADY
GRADED 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] -
(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
160 COMPREHENSIVE QUESTIONS AND ANSWERS | REAL AND
EXACT | 100% RATED CORRECT | 100% VERFIED | ALREADY
GRADED 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] -
(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
