Georgia Institute Of Technology
ISYE 6644
Week 9 Homework - Spring 2021
Due Mar 26 at 11:59pm Points 18 Questions 18
Available Mar 15 at 8am - Mar 26... [Show More] at 11:59pm 12 days Time Limit None
This quiz was locked Mar 26 at 11:59pm.
Attempt History
Attempt Time Score
LATEST Attempt 1 26 minutes 17 out of 18
Score for this quiz: 17 out of 18
Submitted Mar 26 at 11:59pm
This attempt took 26 minutes.
Question 1 1 / 1 pts
(Lesson 7.1: Introduction to Random Variate Generation.) Unif(0,1) PRNs can be used
to generate which of the following random entities?
Correct! Correct! f. All of the above --- and just about anything else!
(f).
Question 2 1 / 1 pts
Week 9 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/165326/quizzes/213873?module_...
1 of 11 4/5/2021, 5:41 PM
(Lesson 7.2: Inverse Transform Theorem --- Intro.) If is an Exp( ) random variable
with c.d.f. , what's the distribution of the random variable ?
X λ
F(x) = 1 - e-λx 1 - e-λX
Correct! Correct! a. Unif(0,1)
Note that , where the last step follows by the
Inverse Transform Theorem. Thus, the correct answer is (a).
1 - e-λX = F(X) ∼ Unif(0, 1) Question 3 1 / 1 pts
(Lesson 7.2: Inverse Transform Theorem --- Intro.) If is a Unif(0,1) random variable,
what's the distribution of ?
Since and are both Unif(0,1) (by symmetry), we have
where the last step follows from Lesson 2's Inverse Transform
Theorem example. Thus, the answer is (d).
U 1 - U
- λ1 ln(U) ∼ - λ1 ln(1 - U) ∼ Exp(λ),
Week 9 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/165326/quizzes/213873?module_...
Question 4 1 / 1 pts
(Lesson 7.2: Inverse Transform Theorem --- Intro.) Suppose that are
i.i.d. Unif(0,1) random variables. Using Excel (or your favorite programming language),
simulate . Draw a histogram of the 5000 numbers.
What p.d.f. does the histogram look like?
Correct! Correct! d. Exponential
By the Inverse Transform Theorem, all of the 's are Exp( ). Since we
have a histogram of 5000 of these, it really ought to look like an exponential
p.d.f., , . Thus, the answer is (d).
Question 5 1 / 1 pts
(Lesson 7.3: Inverse Transform --- Continuous Examples.) Suppose the c.d.f. of is
, . Develop a generator for and demonstrate with .
X
Week 9 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/165326/quizzes/213873?module_...
3 of 11 4/5/2021, 5:41 PM
Set . Then , and so . Plugging
in , we get . Thus, the correct answer is (c)
(Lesson 7.3: Inverse Transform --- Continuous Examples.) If is a Nor(0,1) random
variate, and is the Nor(0,1) c.d.f., what is the distribution of ?
By the Inverse Transform Theorem, Φ(X) ∼ Unif(0, 1); so the answer is (a).
Question 7 1 / 1 pts
(Lesson 7.3: Inverse Transform --- Continuous Examples.) If is a Unif(0,1) random
variate, and is the Nor(0,1) c.d.f., what is the distribution of ?
Week 9 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/165326/quizzes/213873?module_...
4 of 11 4/5/2021, 5:41 PM
By the Inverse Transform Theorem,
Thus, (e) is the correct answer.
Question 8 1 / 1 pts
(Lesson 7.4: Inverse Transform --- Discrete Examples.) How would you simulate the
sum of two 6-sided dice tosses? (Note that is the round-up function; and all of the
's denote PRNs.)
Choice (a) just gives a random real number between 0 and 12. (b) gives a
discrete uniform random integer from 1,2,...,12. (c) gives a continuous triangular
distribution. Meanwhile, recall that we learned in class that is a 6-sided die
toss. Thus, since (d) is simply the sum of two of these tosses, it is the correct
answer.
⌈6U⌉
Question 9 1 / 1 pts
(Lesson 7.4: Inverse Transform --- Discrete Examples.) If is Unif(0,1), how can we
simulate a Geom(0.6) random variate?
5 of 11 4/5/2021, 5:41 PM
Correct! Correct! e. Both (a) and (b)
From the notes, we have
So the answer is (e).
Question 10 1 / 1 pts
(Lesson 7.5: Inverse Transform --- Empirical Distributions.) BONUS: Consider four
observations from some unknown distribution,
, , , and . What is the fourth order statistic, which
we denoted by in class?
X(4)
Correct! Correct! c. 2.7
merely means the largest of the sample of 4 observations. Thus, (c) is the
correct answer.
X(4)
Question 11 1 / 1 pts
(Lesson 7.6: Convolution.) Suppose that and are PRNs. Let . Simulate
this 5000 times, and draw a histogram of the 5000 numbers. What p.d.f. does the
histogram look like?
U V X = U + V
Week 9 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/165326/quizzes/213873?module_...
6 of 11 4/5/2021, 5:41 PM
Correct! Correct! c. Triangular
By the lesson notes, we know that the 5000 Xi's are all Triangular(0,1,2). Since
we have a histogram of 5000 of these, it really ought to look like a triangular
p.d.f. Thus, the answer is (c).
Question 12 1 / 1 pts
(Lesson 7.6: Convolution.) Suppose that are i.i.d. PRNs. What is the
approximate distribution of ?
Correct! Correct! f. Nor(41,18)
7 of 11 4/5/2021, 5:41 PM
By the lesson notes regarding the "desert island'' normal generator, we know that
Thus,
Thus, (f) is the correct answer.
Question 13 1 / 1 pts
(Lesson 7.6: Convolution.) If are PRNs, what's the distribution of
?
U1, U2, U3
-2 ln(U12(1 - U2)2U32)
Correct! Correct! d. Erlang3(1/4)
Since and are both Unif(0,1), we have
Thus, the answer is (d).
U 1 - U
-2 ln(U12(1 - U2)2U32) = -4 ln(U1(1 - U2)U3) ∼ -4 ln(U1U2U3) ∼ Erlang3(1/4).
Question 14 1 / 1 pts
(Lesson 7.7: Acceptance-Rejection --- Intro.) In general, the majorizing function is
itself a p.d.f. [Show Less]
