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Probability Theory MA 381/581 Course Information Fall 2008 Instructor Abbas Alhakim, Ph.D. 361B Science Center Tel. 268-3831 http://people.clarkson.edu/~aalhakim [email protected] Office Hours: 1:00PM--2:30 PM MTW, 12 PM --1 PM on Tuesday, or by appointment. Lectures Monday, Wednesday and Friday 4:00--4:50am SC 362 Textbook:Probability and Statistics (third edition) by DeGroot and Schervish Course Content and Learning Objectives: Catalogue description: Sample spaces; combinatorial analysis, the concept of probability; random variables, expected values; and distributions including hypergeometric, binomial, Poisson, and normal. Details: This course deals with the concepts of probability theory. Starting with the basic axioms that define the subject, we will cover such techniques of calculating probabilities ranging from counting methods to integration, also discussing the meaning and interpretation of the probability of a random outcome. Important concepts include conditional probability, independence of events; random variables: their densities, distributions, distributions of their transformations, their measures of centrality and scale through the study of medians, mathematical expectations, moments, variances and moment generating functions. In particular, we will discuss a collection of random variables that are most widely used in applications (binomial, geometric, hypergeometric, exponential, normal distributions and more). We will also study-mathematically--the law of large numbers (the law of averages) and the Central Limit Theorem. Throughout the course of this discussion examples that relate these theoretical concepts to the world of applications will be provided. Also some [hopefully] entertaining classical problems that shed light on the motivation and objectives of the subject matter will be seen. Detailed objectives of individual lectures will be given in the beginnings of the lecture notes. Graduate students: the content is basically the same, but you are expected to do more work and you will be assigned a few more problems (problems listed with superscript G). Prerequisites and Course Outcomes: Students who intend to take this course should be aware that the content is inherently mathematical. (in fact much more so than Stat 383). The main vehicle for understanding the material is a solid understanding of Calculus: differentiation, integration, infinite series as well as Calculus of two or more variables. Note that some of the essential Calculus concepts will be briefly recalled as needed, however, Calculus I, II and III are all necessary prerequisites. The global objective of the course is to familiarize the students with the concepts of randomness, equip them with the methodology needed to deal with randomness and uncertainty, as encountered in Economics, Engineering or scientific disciplines. This will be accomplished mainly by the examples that illustrate the general theory and by the problems that will be assigned. Academic Integrity: "The Clarkson student will not present, as his or her own, the work of another, or any work that has not been honestly performed, will not take any examination by improper means, and will not aid and abet another in any dishonesty." (Clarkson Regulations) Any student violating this standard will receive an F in the course and will not be allowed to submit any further work. You are allowed, and in fact expected, to work with other students on homework and projects. However, what you turn in for grade must represent your own understanding of the assignment in your own writing style. Exams: There will be three midterm exams in addition to the final. The tentative dates are: Test 1: Friday, October 3th Test 2: Monday, October 27th Test 3: Monday, November 24th Final: Week of December 8 Tests will be given in class during the regular class time. The material to be covered on exams will be outlined at least two classes before the exam. The Final Exam will be comprehensive, covering the whole course. Anyone unable to take an exam should contact me ahead of time and submit a valid excuse in order to be given a make up exam. Any exam missed without prior approval receives a grade of zero. Any appeals to grades should be done in the instructor’s office rather than the classroom. Grading Attendance/class participation/attitude/willingness to learn new things ……………5% Homework……………………………………………………....................................10% Projects…………………………………………………….........................................10% Quizzes………………………………………………………………………………...8% The three midterm Tests(combined)............................................................................42% Final Exam……………………………………………...............................................25% Letter grades will only be given at the end of the term using the weight described above and following the rule: B: 80—84.99% B+: 85—89.99% A: 90 — 100% C: 70—74.99% D: 60—64.99% C+: 75—79.99% D+: 65—69.99% Below 60%: F Homework Suggested homework sets from all the sections that will be covered are listed below. Exercise numbers with a superscript (*) are particularly important and some (or all) of them will be collected for grading as will be announced in due time. Exercises with a superscript (G) are ONLY required for graduate students while every one is encouraged to try them. No late homework will be accepted (unless prior arrangement is made). Quizzes will be given on (some) Fridays and will be announced earlier in the week. There will be no make up quizzes. Projects About three projects will be given during the semester which will typically involve extensive computation/simulation. Every two students must team up to do and submit one project report, (Grad students have to work individually.) The project must be well written with all steps/ideas expressed clearly. The project report must be typed. Homework Sets: 1.4: 1,2*,3,5G,6*,7,8* 1.5: 4, 5, 6*, 7, 8*, 9*, 10, 11*, 12G,13G 1.6: 2*, 4*, 6*, 8* 1.7: 4, 5, 6*, 7, 8*, 9, 10* 1.8: 6, 9, 10*, 12*, 15, 16*, 17, 18, 19 4.1: 3,4*,5,6*,7G,8*,11* 4.2: 2,3*,4*,7*,9* 4.3: 2, 4*,5,6*,8G,9* 4.4: 1,2*,6*,7,12*,13G 4.5: 2*,6a*, 6bG 4.6: 2*,3*,6*,7,9* 4.8: 1,2*,3*,4G,5,6* 1.9: 1, 2*, 3, 4*, 8*, 11G 1.10: 1, 2, 3, 5, 6*, 8*,10* 2.1: 2, 4*,5*,6*,9, 10 2.2: 4, 6*, 8*, 9, 10*, 12, 13, 14* 2.3: 2*, 3, 8*, 10* 2.4G: 2G, 4G, 8G, 10G 2.5G: 1G, 2G, 3G 3.1: 2,4*,5*,7,10*,11* 3.2: 1,3,4,5*,7*,8*,11 3.3: 2*,3,4*,6*,8G,9,14 3.4: 2*,4*,5,7G,8* 3.5: 2*,3,4,6*,8*, ,10G,11G 3.6: 2*,3,4*,5,6,8* 3.8: 2*,3,4*,6*, 7a*,7bG 3.9: page 175: 2*,4*; page 179: 20*,21*, 22*,26G 5.1: 1,2*,3,4*,6*,10* 5.2: 2*,3,4*,5,6*,7 5.3: 1,2*,3,4*,8*,12*,15G 5.4: 1,2*,3*,6*,7,9* 5.6: 1,2*,3*,5*,6*,7*,8G, 17*,18*,20* 5.7: 1,2*,3,4*,6*,7*,9*