Week 8 Homework_ Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
pdf
keyboard_arrow_up
School
Georgia Institute Of Technology *
*We aren’t endorsed by this school
Course
6644
Subject
Industrial Engineering
Date
Jan 9, 2024
Type
Pages
11
Uploaded by KidIronPuppy25
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
1/11
Week 8 Homework
Due
Jul 14 at 11:59pm
Points
10
Questions
10
Available
after Jul 5 at 8am
Time Limit
None
Attempt History
Attempt
Time
Score
LATEST
Attempt 1
19 minutes
10 out of 10
Correct answers will be available on Jul 17 at 12am.
Score for this quiz:
10
out of 10
Submitted Jul 12 at 6:05pm
This attempt took 19 minutes.
1 / 1 pts
Question 1
(Lesson 7.10: Acceptance-Rejection --- Poisson Distribution.) Suppose that
,
, and
. Use our
acceptance-rejection technique from class to generate
.
(You may not need to use all of the uniforms.)
a. N=0
b. N=1
c. N=2
d. N=3
e. N=4
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
2/11
Define
. We'll stop as soon as
.
Let's make the following convenient table.
So we take N = 3, and the answer is (d).
1 / 1 pts
Question 2
(Lesson 7.11: Composition.) BONUS: It's Raining Cats and Dogs is a pet
store with 60% cats and 40% dogs. The weights of cats are Nor(12,4), and
the weights of dogs are Nor(30,25). How would we use composition to
simulate the weight
of a random pet from the store? (Let
denote the
standard normal c.d.f., and let
's denote PRN's.)
a.
b.
c. If
, then
; otherwise,
d. If
, then
; otherwise,
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
3/11
e. If
, then
; otherwise,
By inverse transform, the weight of a cat all by itself is
Similarly, the weight of a dog all by itself is
.
These facts eliminate choices (a), (c), and (e). In addition, (a) and (b)
are some kind of mutant cat-dog, both of which sort of combine 0.6 of
a cat with 0.4 of a dog; so those are wrong. What we really want is to
take
with probability 0.6, and otherwise
. This is
choice (d)!
1 / 1 pts
Question 3
(Lesson 7.12: The Box-Muller Method.) Suppose
and
are i.i.d.
Unif(0,1) with
and
. Use the "cosine" version of Box-
Muller to generate a single Nor(-1,4) random variate. Don't forget to use
radians instead of degrees!
a. -0.326
b. 0
c. 0.326
d. 0.663
e. 1.96
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
4/11
Box-Muller immediately gives the following Nor(0,1) random variate:
To obtain the realization of the Nor($-$1,4), we simply apply the
transform
which is choice (c).
1 / 1 pts
Question 4
(Lesson 7.13: Generating Order Statistics.) Consider i.i.d. Exp(
) random
variables
, and let
. How can we generate
using just
one
PRN?
a.
b.
c.
d.
e.
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
5/11
Let F
denote the Exp(
) c.d.f. Recall that the inverse
is
a fact that we'll use in a minute.
Meanwhile, as also explained in class, the c.d.f.\ of $Y$ is
Now, by inverse transform,
, and so
. This implies that
, and thus
where the last equality
follows from
. This is choice (c).
1 / 1 pts
Question 5
(Lesson 7.14: Multivariate Normal Distribution.) Suppose I have a matrix
. Find the lower triangular matrix
such that
and tell me what the entry
is.
a. -1
b. -0.7071
c. 0
d. 0.7071
e. 1
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
6/11
From our class notes on multivariate normal random variate
generation, we know that the Cholesky matrix we need is
Therefore, the correct answer is (b).
1 / 1 pts
Question 6
(Lesson 7.15: Baby Stochastic Processes.) BONUS: Consider a Markov
chain in which
if it rains on day
; and otherwise,
. Denote
the day-to-day transition probabilities by
Suppose that the probability state transition matrix is
Suppose that it rains on Monday, e.g.,
. Use simulation to find the
probability that it rains on Wednesday, e.g., estimate
.
[You may have to simulate the process a bunch of times in order to estimate
this probability.]
a. 0
b. 0.64
c. 0.72
d. 0.8
e. 1
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
7/11
I'll actually give the analytical solution.
This is answer (c).
1 / 1 pts
Question 7
(Lesson 7.16: Nonhomogeneous Poisson Processes.) Suppose that the
arrival pattern to a parking lot over a certain time period is an NHPP with
. Use simulation to find the probability that there will be exactly 3
arrivals between times
and
.
a. 0
b. 0.195
c. 0.5
d. 0.805
e. 1
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
8/11
I'll give the analytical solution. First of all, the number of arrivals in
that time interval is
Thus,
This is answer (b).
1 / 1 pts
Question 8
(Lesson 7.17: Time Series Generation. ) BONUS: Suppose that
and consider the time series
,
, where the
's are i.i.d. Nor
. (The funny
variance of
guarantees that
for all
). Use simulation to find
. Hint: Simulate
many times. For each run of the
simulation, save the pair
. Then use those pairs to estimate the
covariance.
a. 0
b. 0.7
c. 0.49
d.
e.
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
9/11
By class notes, the analytical answer is
, so that (d) is
correct.
1 / 1 pts
Question 9
(Lesson 7.19: Brownian Motion.) Let
denote a Brownian motion
process at time
. Calculate
.
a. 0
b. 2
c. 3
d. 5
e. 8
We have
, so that the answer
is (c).
1 / 1 pts
Question 10
(Lesson 7.19: Brownian Motion.) BONUS: Let
denote a Brownian
motion process at time
and define a Brownian bridge by
for
. Find the variance of the area under a
bridge, i.e.,
. I'm a nice guy, so I'll get you started...
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
10/11
a. -1/2
b. 0
c. 1/12
d. 1/2
e. 1
Screenshot 2018-03-26 13.46.16.png
(Lesson 7.19: Brownian Motion.) As we discussed in class, you can use
Brownian motion to estimate option prices for stocks. I'm not going to have
you simulate that, but I'm going to give you a quick look-up assignment. As I
write this on May 10, 2019, IBM is currently selling for about $135 per share.
Suppose I'm interested in guaranteeing that I can buy a share of IBM for at
most $145 on Sept. 20, 2019. Look up (maybe using something like
FaceTube on the internets) the corresponding stock option price. [You don't
have to write down an answer for this problem, but I'd like you to do the look-
up anyway.]
As I was writing this solution sheet, the option price was $2.72 --- but this is
obviously subject to change depending on how the market does.
7/12/2019
Week 8 Homework: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN
https://gatech.instructure.com/courses/52126/quizzes/55301
11/11
Quiz Score:
10
out of 10