Week 1 Homework

docx

School

Georgia Institute Of Technology *

*We aren’t endorsed by this school

Course

6242

Subject

Statistics

Date

Feb 20, 2024

Type

docx

Pages

Uploaded by BailiffNeutron10026

Week 1 Homework  Due Jan 19 at 8:59pm  Points 12  Questions 12  Available Jan 12 at 5am - Jan 22 at 8:59pm  Time Limit None Instructions Please answer all the questions below. This quiz was locked Jan 22 at 8:59pm. Attempt History Attempt Time Score LATEST Attempt 1 50 minutes 12 out of 12 Score for this quiz: 12 out of 12 Submitted Jan 12 at 8:51pm This attempt took 50 minutes.

Question 1 1 / 1 pts (Lesson 1.3: Deterministic Model.) Suppose you throw a rock off a cliff having height ℎ 0 = 1000 feet. You're a strong bloke, so the initial downward velocity is �0 = -100 feet/sec (slightly under 70 miles/hr). Further, in this neck of the woods, it turns out there is no friction in the atmosphere - amazing! Now you remember from your Baby Physics class that the height after time � is ℎ (�)= ℎ 0+�0�−16�2 When does the rock hit the ground? a. -11.625 sec b. 2 sec Correct! c. 5.375 sec Set 0= ℎ (�)=1000−100�−16�2, and solve for t. Quadratics are easy: �=−�±�2−4��2�=100±1002+4(16)(1000)2(−16)=−100±272.032=5.375, which we take as the answer since the negative answer doesn't make practical sense. d. 11.625 sec e. 10 sec Set 0= ℎ (�)=1000−100�−16�2, and solve for t. Quadratics are easy: �=−�±�2−4��2�=100±1002+4(16) (1000)2(−16)=−100±272.032=5.375, which we take as the answer since the negative answer doesn't make practical sense.

Question 2 1 / 1 pts (Lesson 1.3: Stochastic Model.) Consider a single-server queueing system where the times between customer arrivals are independent, identically distributed Exp(λ = 2/hr) random variables; and the service times are i.i.d. Exp(µ = 3/hr). Unfortunately, if a potential arriving customer sees that the server is occupied, he gets mad and leaves the system. Thus, the system can have either 0 or 1 customer in it at any time. This is what’s known as an M/M/1/1 queue. If �(�) denotes the probability that a customer is being served at time t, trust me that it can be shown that �(�)=��+�+[�(0)−��+�]�−(�+�)�. If the system is empty at time 0, i.e., �(0)=0, what is the probability that there will be no people in the system at time 1 hr? a. 1 b. 2/3 c. 0.397 Correct! d. 0.603 At time �=1, we have 1−�(�)=1−��+�−[�(0)−��+�]�−(�+�)�=��+�−[�(0)−��+�]�− (�+�)�=32+3−[0−25]�−(5)(1)=0.603 At time �=1, we have 1−�(�)=1−��+�−[�(0)−��+�]�−(�+�)�=��+�− [�(0)−��+�]�−(�+�)�=32+3−[0−25]�−(5)(1)=0.603 Question 3 1 / 1 pts (Lesson 1.4: History.) Harry Markowitz (one of the big wheels in simulation language development) won his Nobel Prize for portfolio theory in 1990, though the work that earned him the award was conducted much earlier in the 1950s. Who won the 1990 Prize with him? You are allowed to look this one up. Correct!

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

a. Merton Miller and William Sharpe for accomplishments in related (but slightly different) subject areas. b. Henry Kissinger c. Albert Einstein d. Subrahmanyan Chandrasekhar for accomplishments in related (but slightly different) subject areas. Question 4 1 / 1 pts (Lesson 1.5: Applications.) Which of the following situations might be good candidates to use simulation? (There may be more than one correct answer.) a. We put $5000 into a savings account paying 2% continuously compounded interest per year, and we are interested in determining the account's value in 5 years. Correct! b. We are interested in investing one half of our portfolio in fixed-interest U.S. bonds and the remaining half in a stock market equity index. We have some information concerning the distribution of stock market returns, but we do not really know what will happen in the market with certainty. Correct! c. We have a new strategy for baseball batting orders, and we would like to know if this strategy beats other commonly used batting orders (e.g., a fast guy bats first, a big, strong guy bats fourth, etc.). We have information on the performance of the various team members, but there’s a lot of randomness in baseball.

d. We have an assembly station in which “customers” (for instance, parts to be manufactured) arrive every 5 minutes exactly and are processed in precisely 4 minutes by a single server. We would like to know how many parts the server can produce in a hour. Correct! e. Consider an assembly station in which parts arrive randomly, with independent exponential interarrival times. There is a single server who can process the parts in a random amount of time that is normally distributed. Moreover, the server takes random breaks every once in a while. We would like to know how big any line is likely to get. Correct! f. Suppose we are interested in determining the number of doctors needed on Friday night at a local emergency room. We need to insure that 90% of patients get treatment within one hour. (a) and (d) do not require simulation, since we can easily “solve” those models with a simple equation or two. (b), (c), (e), and (f) will likely require simulation. Question 5 1 / 1 pts (Lessons 1.6 and 1.7: Baby Examples.) The planet Glubnor has 50-day years. Suppose there are 2 Glubnorians in the room. What’s the probability that they’ll have the same birthday? a. 1/(49 · 50) Correct! b. 1/50 b. Let’s call the two guys A and B. Whatever A’s birthday is, the probability that B matches it is 1/50. Let’s try it another way. The total number of ways that two people can have birthdays is 50 × 50 = 2500. The total number of ways that they can have two different birthdays is 50 × 49 = 2450. Thus, �(��ℎ)=1−�(��ℎ��)=1−24502500=1/50.

c. 1/25 d. 2/49 Let’s call the two guys A and B. Whatever A’s birthday is, the probability that B matches it is 1/50. Let’s try it another way. The total number of ways that two people can have birthdays is 50 × 50 = 2500. The total number of ways that they can have two different birthdays is 50 × 49 = 2450. Thus, �(��ℎ)=1−�(��ℎ��)=1−24502500=1/50. Question 6 1 / 1 pts (Lessons 1.6 and 1.7: Baby Examples.) The planet Glubnor has 50-day years. Now suppose there are 3 Glubnorians in the room. (They’re big, so the room is getting crowded.) What’s the probability that at least two of them have the same birthday? a. 1/50 b. 2/50 c. 1/(49 · 50) Correct! d. 0.0592 d. I admit that this involves a teensy bit of probability (that you will eventually review in Module 2), but it should be easy enough. Mimicking the previous question, we have 1−�(��ℎ��)=1−50(49)(48)503=0.0592. d. I admit that this involves a teensy bit of probability (that you will eventually review in Module 2), but it should be easy enough. Mimicking the previous question, we have

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

1−�(��ℎ��)=1−50(49)(48)503=0.0592. Question 7 1 / 1 pts (Lessons 1.6 and 1.7: Baby Examples.) Inscribe a circle in a unit square and toss �=500 random darts at the square. Suppose that 380 of those darts land in the circle. Using the technology developed in this lesson, what is the resulting estimate for �? a. −3.14 b. 2.82 Correct! c. 3.04 (c), since the estimate �^=4 × (proportion in circle). d. 3.14 e. 3.82 (c), since the estimate �^=4 × (proportion in circle). Question 8 1 / 1 pts (Lessons 1.6 and 1.7: Baby Examples.) Again inscribe a circle in a unit square and toss � random darts at the square. What would our estimate be if we let �→∞ and we applied the same ratio strategy to estimate �? Correct! a. � by the Law of Large Numbers. b. �/2

c. 3.04 d. 3.14 e. 2� (a), by the Law of Large Numbers. Question 9 1 / 1 pts (Lessons 1.6 and 1.7: Baby Examples.) Suppose customers arrive at a single- server ice cream parlor times 3, 6, 15, and 17. Further suppose that it takes the server 7, 9, 6, and 8 minutes, respectively, to serve the four customers. When does customer 4 leave the shoppe? a. 18 b. 25 Correct! c. 33 Here is the sequence of relevant events �� 13371026109193151962541725833 d. 45 (c). Here is the sequence of relevant events �� 13371026109193151962541725833 Question 10 1 / 1 pts

(Lesson 1.8: Generating Randomness.) Suppose we are using the (awful) pseudo-random number generator ��=(5��−1+1)��(8), with starting value ("seed") �0=1. Find the second PRN, �2=�2/�=�2/8. a. 0 b. 1/8 Correct! c. 7/8 We have �1=(5�0+1)��(8)=6��(8)=6��ℎ��2=(5�1+1)��(8)=31�� (8)=7��2=�2/8=7/8 d. 3 We have �1=(5�0+1)��(8)=6��(8)=6��ℎ��2=(5�1+1)��(8)=3 1��(8)=7��2=�2/8=7/8 and the answer is (c). Question 11 1 / 1 pts (Lesson 1.8: Generating Randomness.) Suppose we are using the "decent" pseudo-random number generator ��=16807��−1��(231−1), with seed �0 = 12345678. Find the resulting integer �1. Feel free to use something like Excel if you need to. a. 352515241 b. 16808

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

Correct! c. 1335380034 This is actually not quite so easy as it may seem, since you have to be a little careful not to lose significant digits. We'll learn more about this in Module 6. In any case, �1=16807(12345678)��(231−1)=207493810146��(231−1)=1335380034, where I multiplied the big numbers and took the mod with the help of Excel. d. 12345679 This is actually not quite so easy as it may seem, since you have to be a little careful not to lose significant digits. We'll learn more about this in Module 6. In any case, �1=16807(12345678)��(231−1)=207493810146��(231−1)=13353 80034, where I multiplied the big numbers and took the mod with the help of Excel. Question 12 1 / 1 pts (Lesson 1.8: Generating Randomness.) Suppose that we generate a pseudo- random number � = 0.128. Use this to generate an Exponential (�=1/3) random variate. a. -6.17 Correct! b. 6.17 From the lesson notes, we have �=−(1/�)ℓ�(�)=−3ℓ�(0.128)=6.17. So the answer is (b).Note: It turns out that �=−(1/�)ℓ�(1−�)=−3ℓ�(0.872)=0.411. would also have been an acceptable answer. Can you see why? c. -0.685 d. 0.685

From the lesson notes, we have �=−(1/�)ℓ�(�)=−3ℓ�(0.128)=6.17. So the answer is (b).Note: It turns out that �=−(1/�)ℓ�(1−�)=−3ℓ�(0.872)=0.411. would also have been an acceptable answer. Can you see why? Quiz Score: 12 out of 12

Week 1 Homework

Related Documents