SCROLL DOWN FOR HOMEWORK 5, DUE MONDAY APRIL 30

• The exam covers all the material from the class since the first exam.
• In preparation for the midterm, I have compiled some practice problems.
  • Book:  1.5.2, 4.2.1, 4.2.2, 4.2.3, 4.2.7 (setup only)
  • Q-1.  Create the linear program that will answer question 4.2.19 (p. 264). If you feel inspired, use Mathematica to solve the linear program once you have found it.
  • Q-2.  Perform a sensitivity analysis on the fertilizer example on page 253 of the book and page 112 of the notes. That is, determine the equilibrium cost of both a unit of phosphate and a unit of nitrate. (Hint: Increase the amount of phosphates available and see how much the profit increases.)
  • Q-3.  Determine the system reliability of the following communication system. There are two different methods of communicating; at least one method must succeed in order for the system to succeed. The first method is by an FM radio, which has reliability 75%. The second method is by a satellite radio; both a physical transmitter must transmit the signal (with 90% reliability) AND the satellite dish must retransmit the signal (with 95% reliability) in order for the satellite radio to succeed.
  • Q-4.  Use a computer simulation to approximate the area inside the curve x⁴+y⁴=1. How much larger is this area than the area of the unit circle (x²+y²=1)?
  • Q-5.  Give the definitions of state space and random variable.
• Here are some additional Mathematica questions that would be fair game:
  • Write an If statement that outputs "Heads" with probabilty one third, and "Tails" with probability two thirds.
  • How would you generate a list of 100 trials of Heads and Tails?
  • What does % represent in Mathematica? What would happen if you evaluated it by itself?
• For informational purposes only, here is a copy of a second exam from the past. No guarantees of similarity are assured.

• Read Sections 1.7, 4.1, 4.2, and 4.4. Then answer the following questions.
  • 5-1.  (8 pts) In this problem you will modify the waiting room algorithm from the notes and tutorial in order to better simulate the arrival of patients. Suppose that the doctor determines that patients are more likely to arrive in the first half of the morning (9:00 to 10:29am) than in the second half of the morning (10:30 to 11:59am). Choose arrival probabilities for these two time periods that continues to ensure that the expected number of patients that arrive in any day is 13.5. (Make sure that you justify that your choices ensure this restriction.) Run your simulation 1000 times to determine if your modification increases, decreases, or keeps the same the expected number of patients in the waiting room at noon. Discuss whether the answer you find is what you expected to find.
  • 5-2.  (5 pts) Answer Question 1.7.2 about soccer ball delivery. Set up the problem like we did in class (ordering soccer balls in multiples of twenty) instead of the way the book does (ordering any number of balls).
  • 5-3.  (7 pts) For the following optimization problem, do the following steps. (a) Write down the linear optimization problem in standard form. (b) Draw the feasible region. (c) Find the optimal solution by hand. (d) Use Mathematica to solve the optimization problem.

    A concrete company makes bags of concrete using beach and river sand. Each pound of beach sand costs 2.5 cents and contains 4 units of fine sand, 3 units of coarse sand, and 4 units of gravel. Each pound of river sand costs 6 cents and contains 3 units of fine sand, 6 units of coarse sand, and 12 units of gravel. Determine the cheapest combination of beach and river sand that will produce a bag of concrete, which requires at least 4 units of fine sand, at least 6 units of coarse sand, and at least 10 units of gravel.

• Read Section 5.3A (pp.377–390); you do not need to know about standard deviation.
  • 4-1.  You are playing a card game with a deck of three cards, the Ace of Hearts, the King of Spades, and the Queen of Clubs. You draw one of the three cards at random and call it Card A. Without replacing it, you then draw a second card at random and call it Card B. Let E₁ be the event where the first card is Black. Let E₂ be the event where the second card is an Ace.
    (a) What is the sample space for the set {Card A , Card B} ?
    (b) Determine the probability p(x) for each element x in the sample space.
    (c) Are the events E₁ and E₂ independent events? Explain.
    (d) Suppose you win 4 dollars when Card A is black and lose 5 dollars when Card A is red. What is the expected value of the amount of money you will win when you play this game? Explain what your answer means.
  • 4-2.  Suppose that you are modeling the arrival of patients at a doctor's office in the first five minutes, starting at 9am. Suppose that at every minute (9:00, 9:01, 9:02, 9:03, 9:04), there is a probability of 0.075 that a patient arrives.
    (a) What is the probability that the first patient arrives at 9:02?
    (b) What is the probability that no patients arrive in these first five minutes?
    (c) What is the probability that exactly one patient will arrive in these first five minutes?
    (d) What is the expected number of patients that arrive in these first five minutes? [Hint: Use independence.]
  • 4-3.  This question involves Mathematica.
    (a) Use the method of Notes p. 86 to simulate flipping a fair coin 1000 times; calculate how many heads appear using the Total command. (In the output, do not show all 1000 flips)
    (b) Use a Table command to repeat this experiment 500 times. The result will be a list of 500 numbers, each representing the number of heads that appeared out of 1000. Take the average of this list by using the command Mean.
    (c) Input the list from part (b) into the Histogram command to see a visualization of the 500 trials.
    (d) Discuss what you see in the answers to part (b) and (c) and how they relate to the real-life coin flipping situation.
  • 4-4.  Recall that our waiting room simulation is the following:

    nwait = 0; busy = 0; endTime = 0;
    For[i = 0, i < 180, i++,
       If[endTime == i, busy = 0];
       newPatient = If[RandomReal[] <= 0.075, 1, 0];
       If[newPatient == 1, nwait++];
       If[busy == 0 && nwait > 0, nwait--; busy = 1; endTime = i + 15];];
    nwait

    (a) What goes wrong (if anything) if the simulation is instead: (Here, we have removed the first line of code)

    For[i = 0, i < 180, i++,
       If[endTime == i, busy = 0];
       newPatient = If[RandomReal[] <= 0.075, 1, 0];
       If[newPatient == 1, nwait++];
       If[busy == 0 && nwait > 0, nwait--; busy = 1; endTime = i + 15];]
    nwait

    (b) What goes wrong (if anything) if the simulation is instead: (Here we have reordered some of the lines of code)

    nwait = 0; busy = 0; endTime = 0;
    For[i = 0, i < 180, i++,
       newPatient = If[RandomReal[] <= 0.075, 1, 0];
       If[newPatient == 1, nwait++];
       If[endTime == i, busy = 0];
       If[busy == 0 && nwait > 0, nwait--; busy = 1; endTime = i + 15];]
    nwait

    (c) What goes wrong (if anything) if the simulation is instead: (Here we have a different reordering of the lines of code.)

    nwait = 0; busy = 0; endTime = 0;
    For[i = 0, i < 180, i++,
       If[busy == 0 && nwait > 0, nwait--; busy = 1; endTime = i + 15];
       If[endTime == i, busy = 0];
       newPatient = If[RandomReal[] <= 0.075, 1, 0];
       If[newPatient == 1, nwait++];]
    nwait

• Updated schedule:
  • Q & A Session: Monday, March 12, 2012
  • Exam 1: Wednesday, March 14, 2012
  • Project Topic due by email on Friday, March 16, 2012
• Exam 1 covers all the material from the class up through Wednesday, 3/7/12. (See detail on Class Notes page.)
• In preparation for the midterm, I have compiled some practice problems.
  • Book:  1.4.2 (p. 42), 3.1.5, 3.1.13, 3.1.14 (p. 149), 3.2.3, 3.2.5 (p. 167–168), 3.3.3 (p. 179), 3.4.9 (p. 196)
  • P-1.  Complete all of Problem 3.2.1 (page 167) by hand, with the simplification that the height (h) is proportional to the time (t). [You'll be fitting the data to a line of the form h=ct for some c.]
  • P-2.  What are the steps in the modeling process? What is done at each step?
  • P-3.  Question 3.1.2 (p149). Explain some advantages and disadvantages in complete sentences; give at least four total (advantages + disadvantages).
  • P-4.  The following data is assumed to fit a logarithmic model, y=a+b*ln[x]. Determine the best values for a and b using the least squares criterion. [You may use Mathematica, but that is not required.]

    x	1.1	1.9	2.8	3.8	5.1	6.2	7.2	8.0	8.7	9.8
    y	3.2	5.2	6.4	7.3	8.0	8.6	9.1	9.5	9.9	10.2

• Here are some Mathematica questions that would be fair game:
  • Explain the difference between the following two lines of Mathematica code:
    a=Table[3i,{i,1,5}];
    a=Table[3i,{i,1,5}]
  • Write down the Mathematica code which would determine the numerical value of the expression Sqrt[10]+Log[2].
  • What are the differences between (), [], and {}?
  • What do you expect when you evaluate the code Table[2^k,{k,1,10}]?
  • When would you use the TableForm command?
  • Explain the difference between the Fit and the FindFit commands.
  • You need to know how to plot a list, how to plot a function, and how to incorporate two plots together.
• For informational purposes only, here is a copy of a first exam from the past. No guarantees of similarity are assured.

• Read Sections 3.2 and 3.4. Then answer the following three problems.
  • 3-1.  Find by hand the least squares best fit line to the data {(3, 8), (5, 14), (7, 23)}.
  • 3-2.  Read http://eagereyes.org/criticism/anscombes-quartet. This question is to understand the set of four figures in the middle of the page. Write three paragraphs explaining this blog post in the context of this class. Explain what makes the four data sets similar; why are they grouped together? Discuss the differences in the figures; does the "line of best fit" fit one of the sets of data better than another? What does "best fit" mean in this context? Last, given these four figures and the lines of best fit, how would you modify your modeling approach to find a better fit to each data set (if necessary).
  • 3-3.  (10 pts) Let's work with some real world data! I have imported some of the data from this website into Mathematica: http://archive.ics.uci.edu/ml/datasets/Housing. Download the Mathematica notebook here. As you can read, this is a data set that contains housing prices and various other factors from different neighborhoods in Boston. We would like to determine the best predictor of housing prices in terms of the thirteen given variables.
            (a) Use Mathematica to generate the scatterplots for each of the 13 factors versus home prices (Column 14). Print these out. Determine the $R^2$ coefficient for the best fit line for each of these scatterplots. Which two or three factors are the most linearly correlated with home prices? Which two or three factors are least linearly correlated with home prices? Do your conclusions make real-life sense?
            (b) Run a multiple linear regression using at least two of the thirteen factors. Show that the $R^2$ value for this multiple linear regression is larger than either single linear regression.
            (c) One of the factors appears to be correlated with house prices in a non-linear fashion. Try to fit the data to a different function that makes sense. Display the best fit curve along with the data.

• Reminder: On Wednesday, February 15, 2012, we will meet in the computer lab.
• Reminder: Tuesday, February 21, 2012 follows a Monday schedule and we have class.
• Read Section 1.3 from page 23 to page 25 and Section 3.2 from page 150 to the middle of page 154.
  • 2-1.  Here is some data that should fit a polynomial of the form \(y=Cx^k\) for some constants \(C\) and \(k\), where \(x\) is an independent variable and \(y\) is a dependent variable. Use the method involving logarithms to approximate \(C\) and \(k\). Explain the steps you applied and how you found your answer.

    $x$	0.5	1	2	4	5	6	8	9	10
    $y$	133	175	256	351	405	443	499	535	574

    ListPlot[Ln[{{0.5, 133}, {1, 175}, {2, 256}, {4, 351}, {5, 405}, {6, 443}, {8, 499}, {9, 535}, {10, 574}}]]
  • 2-2.  You know how to use logarithms to transform data to fit it to a polynomial of the form $y=Cx^k$. (You just did it in Problem 2-1.) It is also possible to use logarithms to transform data to fit data to an exponential function of the form $y=Ce^{kx}$. Use algebra to figure out the necessary transformation. Then describe the steps that you would use to find the constants of best fit $C$ and $k$ if you are given a set of data (such as that in Problem 2-1).
  • 2-3.  (10 pts total)
    (a) Go online to Wolfram.com Broadcast. Watch the video "Mathematica in Action" under "Introductory >>". Then go to Hands On Start. Watch the first two webcasts. The first is 4.5 minutes long and introduces you to how to add sections and into your notebooks. The second is 10 minutes long and discusses how Mathematica 8 allows you to do "free-form input" (but that to use the full capability, you need to understand the Mathematical language) and other ways to input information.
    (b) Create a new Mathematica notebook that satisfies the following properties.
    • Your notebook should have a Title (with relevant title), Subtitle (Your name, date, class). Change the stylesheet (Format > Stylesheet) to your liking.
    • Name a first Section "List Questions" and create two subsections, named as you wish. The two subsections should answer the following two questions. Every question should be answered completely and must include text cells that explain what you are doing.
      • Use the Table command to create a list of the square numbers that are between 100 and 1000 and a list of the cube numbers that are between 100 and 1000.
      • Define variables list1 and list2, each with 20 entries. Have Mathematica count the number of entries in each list and also find the sum of the entries in each list. Then have Mathematica generate a new list named list3 that combines the two lists into a set of twenty ordered pairs. For example, if list1 starts off {1,2,3,..., and list2 starts off {4,5,6,..., then list3 should start off {{1,4},{2,5},{3,6},...}.
    • Name a second Section "Plotting Questions" and create subsections as necessary to answer the following question.
      • Generate three three individual plots and combine them in a fourth plot, as follows. First, create a plot of the sine function from $0$ to $2\pi$ that has a thick green line. Next, create a plot of the cosine function from $0$ to $2\pi$ that has a thin dashed line and is not green. Third, create a plot of ten to twenty large bright fluorescent points of the form $(x,x^2-2x)$ where $x$ ranges from $0$ to $2\pi$. Last, combine all three graphs on one set of axes.
    • Name a last Section "Neat Mathematica things"
      • Find two neat things that you have seen that Mathematica is able to do. Copy the code into your notebook and describe what it does. Explain where you found these things and discuss why they are interesting. Further discuss in what ways you or someone else might modify these things in a new and innovative way.

• Thoroughly read the class web page including the syllabus and schedule. This should answer all the questions that you may have about the class.
• Don't forget to print out the next set of notes for Monday's class.
• Read Sections 1.1 and 1.2 to page 18. Then complete the following problems.
• Problem 1-1 must be completed online before class on Monday 2/6 for credit.
  • 1-1.  (a) Email me at chanusa@qc.cuny.edu. In your message, include your name, your class (Math 245), the email address where you are best contacted, and your anticipated graduation year. Thanks!
            (b) Take the syllabus quiz on Blackboard. Retake the quiz as necessary to earn a score of 100%.
• Problems 1-2 through 1-5 1-4 should be written up and handed in in class before class starts on Monday:
  • 1-2.  Read Section 1.1 from the textbook (pp. 1–12); it gives six different models that describe real-life situations. Write two to three paragraphs explaining where you have experience with mathematical models in real life. (Do not use the examples from the book unless you feel you have a unique perspective.) Explain how the real-life situation and the model differ. (You may want to address the following questions: Are there simplifying assumptions? Does the model truly describe what happens in real life or are there limits to the model's effectiveness? ... )
  • 1-3.  Here are three different vague scenarios. Choose one of the three and identify a precise problem statement that you would like to study. Determine eight variables that affect your proposed problem. Then of those eight variables, choose those that are the most important, and why you think they are more important than the rest.
    1. The Bronx Zoo is planning to increase its profit during next year's "Boo in the Zoo" event.
    2. Your favorite supermarket is planning to reorganize its produce section.
    3. New York City is planning to change the traffic light programming in your neighborhood to better serve the community.
  • 1-4.  How would the model in Galileo's gravity example change if the proportionality assumption (1$\!^*$) on page 17 is replaced by a simple linearity assumption? Discuss the change in the mathematical manipulation step of the modeling process, and explain when this would be a reasonable assumption.
  • 1-5.  Homework 1-5 has been removed.
• Remember what is expected:
  • Write your solutions using complete sentences.
  • Follow the guidelines for turning in homework.
• Important: We will be in the computer lab in Kiely 236 on Monday, February 6 Wednesday, February 8.
  • If you want installing the mathematical software Mathematica on your laptop, bring it to class on Wednesday.

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