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Week 4 Homework
Due Feb 13 at 11:59pm Points 13 Questions 13 Available Feb 6 at 8am - Feb 16 at 11:59pm Time Limit None
This quiz was locked Feb 16 at 11:59pm.
Attempt History
Attempt Time Score LATEST Attempt 1 3,006 minutes 13 out of 13
Score for this quiz: 13 out of 13 Submitted Feb 12 at 2:33am This attempt took 3,006 minutes. Correct answer Question 1 1 / 1 pts
a. 1 b. 2. c. 7. d. 14.
We have
So using h = 0.01, we have
Thus, the answer is (d).
We have
(Lesson 3.1: Solving a Differential Equation.) Suppose that. We know that if is small, then
Using this expression with , find an approximate value for.
So using h = 0.01, we have
Thus, the answer is (d). Correct answer
Question 2 1 / 1 pts
a. b. c. d.
and thus the answer is (d).
e.
and thus the answer is (d).
Correct answer
Question 3 1 / 1 pts
a. b. c.
This takes a little work. The good news is that you can actually get the true answer using the technique of separation of variables. We have
so that
Which implies
(Lesson 3.1: Solving a Differential Equation.) Suppose that. What is the actual value of ?
(Lesson 3.1: Solving a Differential Equation.) Consider the differential equation with. What is the exact formula for?
Wow, what a good match! In any case, the answer is (c).
d.
By previous question, the true answer is the answer is.
But our job is to use Euler to come up with an iterative approximation, so here it goes. As usual, we start with from which we obtain the following table.
Wow, what a good match! In any case, the answer is (c). Correct answer Question 5 1 / 1 pts
a.
In the notation of the lesson, the general approximation we've been using is
(Lesson 3.2: Monte Carlo Integration.) Suppose that we want to use Monte Carlo integration to
approximate. If are i.i.d. Unif(0,1)'s, what's a good approximation
for?
a. 0 b. 0. c. 0. d. 0. , so the answer is (d). e. 0. , so the answer is (d).
Correct answer
Question 7 1 / 1 pts
a. 0. b. 0. Thus, the answer is (b). c. 1. d. 2. Thus, the answer is (b).
Correct answer
Question 8 1 / 1 pts
a. b. 4.0 (UGA answer) c. 3.
(Lesson 3.2: Monte Carlo Integration.) Again suppose that we want to use Monte Carlo integration to
approximate. You may have recently discovered that the MC estimator is of the form
Estimate the integral by calculating with the following 4 uniforms:
(Lesson 3.2: Monte Carlo Integration.) Yet again suppose that we want to use Monte Carlo integration
to approximate. What is the exact value of?
(Lesson 3.3: Making Some .) Inscribe a circle in a unit square and toss random darts at the square. Suppose that 760 of those darts land in the circle. Using the technology developed in class, what is the resulting estimate for?
d. 3. The estimate Thus, the answer is (d).
e. 3.
The estimate Thus, the answer is (d). Correct answer
Question 9 1 / 1 pts
a. 3 b. 9 c. 13 d. 17 Let's make a version of our usual table.
Thus, the answer is (d).
e. 19
Let's make a version of our usual table.
Thus, the answer is (d).
Correct answer
Question 10 1 / 1 pts
a.
(Lesson 3.4: Single-Server Queue.) Consider a single-server Q with LIFO ( last -in-first-out) services. Suppose that three customers show up at times 5, 6, and 8, and that they all have service times of 4. When does customer 2 leave the system?
(Lesson 3.5: Inventory Model.) Consider our numerical example from the lesson. What would the third day's total profits have been if we had used a (4,10) policy instead of a (3,10)?
By the Inverse Transform Theorem, we know that. But since and
are both Unif(0,1) (why?), we also have
In particular,
so that the answer is (b). Correct answer
Question 12 1 / 1 pts
a. Unif(0,2) b. Normal c. Exponential d. Triangular By any of the hints, you get a Triangular(0,1,2) distribution, i.e., answer (d).
By any of the hints, you get a Triangular(0,1,2) distribution, i.e., answer (d).
Correct answer
Question 13 1 / 1 pts
(Lesson 3.6: Simulating Random Variables.) If and are i.i.d. Unif(0,1) random variables, what is the distribution of? Hints: (i) I may have mentioned this in class at some point; (ii) You may be able to reason this out by looking at the distribution of the sum of two dice tosses; or (iii) You can use something like Excel to simulate many times and make a histogram of the results.
(Lesson 3.7: Spreadsheet Simulation.) I stole this problem from the Banks, Carson, Nelson and Nicol text (5th edition). Expenses for Joey's college attendance next year are as follows (in $):
Tuition = 8400 Dormitory = 5400 Meals Unif(900,1350) Entertainment Unif(600,1200) Transportation Unif(200,600) Books Unif(400,800)
Here are the income streams the student has for next year: Scholarship = 3000 Parents = 4000
a. $ b. $ c. $
An easy spreadsheet simulation (or an almost-as-easy exact analytical calculation) reveals that the expected loan amount is $3325, or answer (c).
If you don't believe me, here's some Matlab code (if you happen to have Matlab)...
d. $ e. $
An easy spreadsheet simulation (or an almost-as-easy exact analytical calculation) reveals that the expected loan amount is $3325, or answer (c).
If you don't believe me, here's some Matlab code (if you happen to have Matlab)...
Quiz Score: 13 out of 13
Waiting Tables Unif(3000,5000) Library Job Unif(2000,3000)
Use Monte Carlo simulation to estimate the expected value of the loan that will be needed to enable Joey to go to college next year.
