PRACTICE FINAL ISYE6644 EXAM 2025
QUESTIONS AND 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] - ANS 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. - ANS 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. - ANS 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.). - ANS 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! - ANS TRUE
TRUE or FALSE? Arena has a built-in Input Analyzer tool that allows for the fitting of certain
distributions to data. - ANS TRUE
1 @COPYRIGHT THEBRIGHT 2025/2026
,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. -
ANS 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. - ANS 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. - ANS 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? -
ANS 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. - ANS 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. - ANS 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.) - ANS 0.854
2 @COPYRIGHT THEBRIGHT 2025/2026
, 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. - ANS 0.1587
Which one of the following properties of a Brownian motion process W(t) is FALSE? -
ANS W(3) − W(1) is independent of W(4) − W(2).
Find the sample variance of −10, 10, 0. - ANS 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 ? - ANS 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 - ANS 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 ? - ANS 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). - ANS 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
3 @COPYRIGHT THEBRIGHT 2025/2026
QUESTIONS AND 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] - ANS 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. - ANS 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. - ANS 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.). - ANS 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! - ANS TRUE
TRUE or FALSE? Arena has a built-in Input Analyzer tool that allows for the fitting of certain
distributions to data. - ANS TRUE
1 @COPYRIGHT THEBRIGHT 2025/2026
,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. -
ANS 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. - ANS 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. - ANS 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? -
ANS 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. - ANS 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. - ANS 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.) - ANS 0.854
2 @COPYRIGHT THEBRIGHT 2025/2026
, 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. - ANS 0.1587
Which one of the following properties of a Brownian motion process W(t) is FALSE? -
ANS W(3) − W(1) is independent of W(4) − W(2).
Find the sample variance of −10, 10, 0. - ANS 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 ? - ANS 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 - ANS 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 ? - ANS 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). - ANS 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
3 @COPYRIGHT THEBRIGHT 2025/2026
