Lawrence Technological UniversityMSIE EMS5603 Week 8 Homework _ Simulation - ISYE-6644-OAN_O01Due Mar 8 at 11:59pm Points 12 Questions 12Available after Feb 28 at 8am Time Limit NoneAħempt HistoryAttempt Time ScoreLATEST Attempt 1 10,509 minutes 11 out of 12Score for this quiz: 11 out of 12Submitted Mar 8 at 11:59pmThis attempt took 10,509 minutes.Question 1 1 / 1 pts(Lesson 6.1: Introduction t
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MSIE EMS5603 Week 8 Homework _ Simulation - ISYE-6644-OAN_O01
Due Mar 8 at 11:59pm Points 12 Questions 12
Available after Feb 28 at 8am Time Limit None
Aħempt History
Attempt Time Score
LATEST Attempt 1 10,509 minutes 11 out of 12
Score for this quiz: 11 out of 12
Submitted Mar 8 at 11:59pm
This attempt took 10,509 minutes.
Question 1 1 / 1 pts
(Lesson 6.1: Introduction to Uniform Random Numbers.) TRUE or
FALSE? It's actually a good thing for you to be able to reproduce a
sequence of PRNs, should you so desire.
True
False
TRUE. [You'd better to be able to reproduce your work if your boss
doesn't believe you!]
Question 2 1 / 1 pts
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(Lesson 6.2: Some Lousy Generators.) Consider von Neumann's midsquare PRN method, and suppose that . What is
? [Note: For purposes of this problem, treat all 's as
if they were 8 digits, e.g., treat 123456 as if it were 00123456.]
a. 9834
b. 0.9834
c. 50055625
d. 556
e. 0.0556
Following the notes, , so that , so that
. Similarly, and , so that
. Thus, (e) is the correct answer.
Question 3 1 / 1 pts
(Lesson 6.3: Linear Congruential Generators.) YES or NO? Does
have full period?
True
False
YES! [It trivially cycles through .]
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Question 4 1 / 1 pts
(Lesson 6.3: Linear Congruential Generators. Problem 7.1 from Law
2015). Consider the generator
.
Starting from , find .
a. 0
b. 6
c. 7
d. 11
e. 38
If , then , , , ...,
. So the cycle repeats itself every 16 iterations. This
implies that , and so
. Thus, the answer is (d).
Question 5 1 / 1 pts
(Lesson 6.3: Linear Congruential Generators.) Which uniform generator
was recommended in class, at least as a ``desert island'' generator?
a.
b.
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c.
d.
(b).
Question 6 1 / 1 pts
(Lesson 6.4: Tausworthe Generators.) Suppose that a Tausworthe
generator gave you the series of bits 1010101. If you use all 7 bits, what
Unif(0,1) random number would that translate to?
a. 0.3825
b. 0.5
c. 0.6641
d. 0.9826
(c). Using the usual base-2 notation, we have
Question 7 1 / 1 pts
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(Lesson 6.5: Generalizations of LCGs.) TRUE or FALSE? There are
some great PRN generators out there with incredible cycle lengths
and even !
True
False
TRUE! Amazing!
Question 8 1 / 1 pts
(Lesson 6.6: Choosing a Generator --- Theory.) Which of the following
statements about the RANDU generator is true?
a. Something just ain't right about that boy.
b. The generator is given by
c. The PRNs appear at first glance to be uniform, but funny things happen
when you look at the plots of the PRNs in multiple dimensions.
d. The PRNs are distributed on just 15 hyperplanes.
e. All of the above.
(e).
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Question 9 1 / 1 pts
(Lesson 6.7: Statistical Considerations - Intro.) Suppose the guy on trial is
actually guilty but you incorrectly acquit him. So you've incorrectly
accepted the null hypothesis of innocence. What type of error have you
just made - Type I or Type II?
a. Type I
b. Type II
(b). You've incorrectly accepted the null hypothesis of innocence.
Question 10 1 / 1 pts
(Lesson 6.8: Goodness-of-Fit Tests.) Suppose we observe 1000 PRNs to
obtain the following data.
Conduct a goodness-of-fit test to see if these numbers are
approximately Unif(0,1). Use level of significance .
Here are some table entries that you may need: ,
, and . ACCEPT or REJECT?
a. Accept
b. Reject
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The expected number of observations per equal-probability cell is
. This gives us the following
augmented table.
Thus, the goodness-of-fit statistic is
Meanwhile, the appropriate quantile .
Since , we ACCEPT . In other words, we are willing
to assume that the numbers are approximately Unif(0,1).
Question 11 1 / 1 pts
(Lesson 6.9: Runs Tests for Independence.) Consider the following
PRNs. (Read from left to right, and then down.)
Let's conduct a runs up and down test to test : the 's are
independent with level = 0.05. ACCEPT or REJECT?
a. Accept
b. Reject
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Question 12 0 / 1 pts
(Lesson 6.9: Runs Tests for Independence.) Suppose that are
i.i.d. Unif(0,1). Let's denote the number of runs up-and-down by . Find
the EXACT distribution of .
a. ~ Unif(0,1)
ou Answered ou Answered b. ~ Norm(0,1)
c.
d.
orrect Answer orrect Answer e.
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