1
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Practice Final Exams ISYE6644 Questions
and Answers (100% Correct Answers)
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]—Ans: TRUE
© 2025 Assignment Expert
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
Guru01 - Stuvia
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
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
, 2
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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
© 2025 Assignment Expert
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
Guru01 - Stuvia
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
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
, 3
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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]
© 2025 Assignment Expert
= 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
Guru01 - Stuvia
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 significance α = 0.05 for your test.—Ans: 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.—Ans: False
For Expert help and assignment solutions, +254707240657
Practice Final Exams ISYE6644 Questions
and Answers (100% Correct Answers)
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]—Ans: TRUE
© 2025 Assignment Expert
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
Guru01 - Stuvia
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
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
, 2
For Expert help and assignment solutions, +254707240657
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
© 2025 Assignment Expert
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
Guru01 - Stuvia
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
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
, 3
For Expert help and assignment solutions, +254707240657
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]
© 2025 Assignment Expert
= 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
Guru01 - Stuvia
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 significance α = 0.05 for your test.—Ans: 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.—Ans: False
