CURRENTLY TESTING SOLUTIONS Practice Final Exams
ISYE6644 WITH ACTUAL CORRECT DETAILED 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] - 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
Which one of the following properties of a Brownian motion process W(t) is FALSE? - ANSWER-W(3) −
W(1) is independent of W(4) − W(2).
Find the sample variance of −10, 10, 0. - 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 ? - 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 - 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 ? - 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). - 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. - 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. - 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? - 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. - ANSWER-[8.5,
16.1]
