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Duke CPS 111 - Homework 5

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CPS111- Homework 5Due on February 21, 2008Questions may continue on the back. Please write clearly. What I cannot read, I will not grade. Typed homework ispreferable. A good compromise is to type the words and write the math by hand.1. Consider the experiment of tossing a single coin once. Let p be the probability of the event E that the coin lands on head.Let F denote the event that the coin lands on tail. Find all the values of p for which the two events E and F are independent.Explain your answer.2. [Make sure you understand the distinction between universe and event space before you answer this question.]Consider the experiment of tossing a coin twice. Let HT denote the outcome in which the first coin toss lands on head,and the second lands on tail. Denote other outcomes with similar notation.(a) Specify the universe and event space for this experiment, and state the number of elements in the event space.(b) Two events are disjoint when their intersection is empty. Are the singleton events {HT } and {T T } in the experimentabove disjoint?(c) Assume that the coin is fair. That is, all outcomes have equal probability. Determine whether the two events E ={T H, T T } and F = {HT, T H} are dependent or independent. Do so in three different ways, involving respectivelyP (E ∩ F ), P (E | F ), and P (F | E).(d) Ditto for E = {T H, T T } and F = {HH, HT, T H}.(e) In the same event space, let {H?} denote the event in which the first toss of the coin results in head, regardless of whathappens in the second toss. Which set from the event space you defined corresponds to {H?}?(f) Show that for a fair coin the two events {H?} and {T ?} (defined analogously) are dependent. What does this mean,intuitively?(g) Let {?T } be the event that the second toss results in a tail (regardless of what happens in the first toss). Show that thetwo events {H?} and {?T } are independent. What does this mean, intuitively?3. This problem invites you to think about possible projects for this class. The purpose of this exercise is to ease youinto a habit of thinking how real-world problems can be formalized. What you say in your answer will not be taken as acommitment to an actual project plan.Write a proposal for a modeling project for this class that uses a deterministic, discrete dynamical system as its mainformalization tool. Your proposal should describe your specific aims (what is the outcome of your work, and what formatit will be delivered in: a report on experiments, a piece of software code,...); some background information (where does theproblem arise, in what way could a model like yours be used to address a specific real-world problem); an outline of theactual model (main inputs, outputs, state vector, and a sketch of the system equations); and a research plan and schedule(what needs to be done, and when; what simplifying assumptions are made; where do data come from; what questions shouldbe addressed with the experiments; how could the model be extended).Rather than giving detailed instructions on how to write your plan, an example is given on the next pages. You may notuse the same model as in the example below for your project plan in this assignment. However, if you like the project ideaproposed here, or a variant of it, you are free to use that for your actual project.A project plan that is on a topic different from the one in the example below, and is described at an equivalent level ofdetail and clarity, will earn full credit for this problem. The exact format (section headers, and so forth) is up to you. Pleaseconsult the following web page for ways to cite sources properly:http://library.duke.edu/research/citing/within/index.htmlCPS111 — Duke — February 13, 2008U.S. Population Growth(Your Name Here)Specific AimsThis project will model U.S. population growth over a given period of time as a deterministic, discrete dynamic system.Input data on population numbers, birth and death rates, and immigration will be inferred from reports by the U.S. CensusBureau [1].The system will use Census population data to initialize the model in year 0 (say, 1900). For subsequent years, onlybirth, death, and immigration rates will be used, not actual population numbers. The latter will be used for validating themodel, that is, for comparing its predictions to known, real data about the past.The final outcome of the project will be a Matlab software package that reads relevant data from a file and produces atable and plots of predictions for a given time period. The package will be accompanied by a brief instruction manual, andby a report that describes the model, the software, and sample predictions. The report will also describe results obtainedduring model validation, and discuss strengths and weaknesses of the model.SignificancePredicting the future numbers of individuals in each of several age groups in a Country is obviously an important source forpolicy planning at all levels of government and for many private organizations. Governments can use these predictions to planfor resources ranging from energy consumption and highway traffic to hospital needs, and from educational infrastructureto tax revenue and social security expenses. Private corporations can predict future markets or availability of workforce inappropriate age categories.For these predictions, it is important to know not just the total number of people in the Country of interest, but also thenumber of people in each age group, and possibly gender, race, or national origin.The ModelThe U.S. Census Bureau [1] divides population into eight age groups. Correspondingly, a state vector x(n) is defined foryears n = 0, . . . , N, where year n = 0 is the initial year of a simulation, and N is the number of years considered in thesimulation. This vector has the following eight components, expressed in number of individuals:x1(n) = Population below age 5 in year nx2(n) = Population of age 5 ≤ a < 15 in year nx3(n) = Population of age 15 ≤ a < 25 in year nx4(n) = Population of age 25 ≤ a < 35 in year nx5(n) = Population of age 35 ≤ a < 45 in year nx6(n) = Population of age 45 ≤ a < 55 in year nx7(n) = Population of age 55 ≤ a < 65 in year nx8(n) = Population of age 65 ≤ a in year nFor year n = 0, data from the U.S. Census Bureau are used to provide a vector x0such thatx(0) = x0.CPS111 — Duke — February 13, 2008For subsequent years, the following parameters are derived from U.S. Census data (where a parameter k is specified,


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