MATH 440: Probability Theory – Fall Term, 2009

 

“The most important questions of life are, for the most part, really only problems of probability.”

— Pierre Laplace (1749 – 1827)

 

Course Goals

Probability theory is an area of mathematics with many fun problems and connections between concepts (and lots of reminders of calculus). I use different textbooks for Probability Theory (MATH 440) and Mathematical Statistics (MATH 445), because even though 440 is a pre-requisite for 445, it is itself a stand-alone course (so I prefer to use a book written by a probabilist, rather than a statistician). Statistical theory uses probability as its underlying mathematical structure (some think statistics is a sub-area of mathematics, but I think statistics is its own science, which borrows some tools from mathematics). So knowledge of probability is necessary in order to understand the theoretical basis of statistics, but probability in itself is an interesting, rich topic. When appropriate I will point out the connection between probability and statistics, but I teach this course to showcase the probability topics (which I think are interesting and connected, yet don’t necessarily have lots of practical applications).

 

Professor Contact Information

Joy Jordan, Associate Professor of Statistics, 410 Briggs Hall  

PHONE: 832-6894, EMAIL: joy.jordan@lawrence.edu, WEB: www.lawrence.edu/fast/jordanj/

 

Please note the URL for my homepage. On this page is a link to the Math 440 web page, where I will post, for example, homework assignments and handouts. I check email regularly (2-3 times a day), but not obsessively. If you need to contact me urgently (e.g., you have a family emergency, you want to make an appointment as soon as possible), then please call me.

 

Required Textbook and Additional Reading

A First Course in Probability, 8th Edition, Ross, 2010, Prentice Hall

 

Sorry, but we must use this new edition, as the 7th edition is not readily available (I try to use old editions—to save students money—when possible, but it wasn’t possible this time). A copy of the textbook is on 2-hour reserve at the library.

 

Additional reading (e.g., journal articles) will be provided throughout the term; this reading is as important as the textbook reading.

 

Office Hours

Monday: 3:00 – 4:30, Tuesday: 1:30 –2:30, Wednesday: 8:30 – 9:30, Thursday: 1:30 – 3:00

 

If these times do not work with your particular class schedule, I am happy to make individual appointments for other times. (You need not make an appointment during regular office hours—just come in.) Please ask if you need help, and I will do all I can to assist you. That said, I expect you to come to office hours prepared (e.g., having done the reading, knowing the definitions) and not simply looking for easy answers. Besides office hours, anytime my door is open, feel free to come in and ask questions. If my door is closed, I am either out of the office, or I’m working and prefer not to be disturbed.

 

Class Participation/Discussion

Class discussion is part of this course, and it will be factored into your final grade. Throughout the term, additional reading will be assigned, and you are responsible for coming to class prepared to effectively and critically discuss this material. Occasionally an individual student will be asked to explain a problem-solution to the whole class. Note that participation also includes asking thoughtful questions in class and office hours, and answering my queries in class (regardless of whether you have the “right” answer).

 

Exams

There will one in-class exam during the term and a final exam. The first exam is on Monday, October 12 and the final exam is Friday, November 20 at 6:30 p.m. (ugh! But we all must get used to this new calendar).
Homework (and the Honor Code)

You will turn in regular homework assignments; these problems (or a subset of the problems) will be graded. There will be two types of homework problems: 1) problems on which you are free to work together and 2) work-alone problems. On the work-together problems, your write-up should be your own (you can talk with other students about the problems, but you must write up the solutions individually—that is, you can write scratch work from your study sessions, but you must use your own words and explanations when writing up your final solution). On the work-alone problems, you should not talk with other students at all (except perhaps to clarify what a question is asking)—please see me in office hours with questions about the work-alone problems. On the assignments, I will clearly differentiate between the two types of problems.

 

Your grade will depend on both the content and exposition of your answers; write up the problems as carefully as you did—or should have done—in Math 310. That is, be sure your logic is clear, your solution reads smoothly (even if using symbols), and that one of your peers could read and understand your solution without asking any additional questions.

 

I take the Lawrence Honor Code very seriously, and recently I’ve been deeply disappointed to hear of more frequent violations of this code. I think the Honor Code is a special quality of the Lawrence experience; a quality that translates beautifully into a life-long behavior. If you are feeling particularly stressed in this class, come talk with me, but don’t violate the Honor Code—I will pursue any case I feel is an honor code violation.

 

Grading

Your final grade is based on a weighting of class participation (10%), work-together homework (25%), work-alone homework (25%), and exams (first exam – 20%, second exam – 20%). The letter grades will be assigned as follows, corresponding to Lawrence’s GPA system (note: the cutoff is the lowest percentage that receives that letter grade):

 

Cutoff

Grade

93.75

A

90.00

A-

86.25

B+

83.75

B

80.00

B-

76.25

C+

73.75

C

70.00

C-

66.25

D+

63.75

D

60.00

D-

 

Coverage

A tentative course schedule is included with this syllabus. We are scheduled to cover the material in Chapters 1 – 7 (and part of 8) of the textbook, as well as additional reading. We may cover more or less material, though, depending on how the course unfolds. I want to be flexible with the schedule in case changes are needed to best encourage learning. (I’ll keep you posted on major changes to the schedule.)

 

Life Balance

Because I love probability and statistics so much, I will encourage you to work hard to learn the material. But please realize that your self-worth is not associated with your letter grade on a particular homework or exam (or even with your final course grade). You are all good people, regardless of your official class performance on tasks.

 

Furthermore, I think as a society in general, and at Lawrence in particular, we are over-scheduled and allow precious little downtime and quiet reflection. I encourage you to think carefully about the intensity and number of courses, activities, and obligations in your life, and to seek balance as much as possible. (I’m happy to talk with you more about this—that is, we can discuss life as well as statistics.)

Tentative Course Schedule

 

Date

General Material

Corresponding Textbook Reading

M 9/14

Introduction to the course and group problem-solving

 

W 9/16

Review: Basic principle of counting, permutations, combinations, binomial and multinomial theorems (group-work)

Sections 1.1 – 1.5  

F 9/18

Discuss problems from Wednesday (student presentations of solutions) and start probability

Sections 2.1 – 2.3

M 9/21

Probability propositions and determining probabilities for equally-likely outcomes

Sections 2.4 – 2.5

W 9/23

Conditional probability, multiplication rule, law of total probability, and Bayes’ rule

Sections 3.1 – 3.3

F 9/25

Bayes’ rule and independence

Sections 3.3, 3.4 (read 3.5 on own)

M 9/28

Finish independence and start discrete random variables

Sections 3.4, 4.1 – 4.2

W 9/30

Expected value and properties of expected value

Sections 4.3 – 4.5

F 10/2

Bernoulli and binomial distributions

Section 4.6

M 10/5

Poisson and hypergeometric distribution (and relationships to the binomial distribution)

Sections 4.7, 4.8.3

W 10/7

Geometric and hypergeometric (briefly) distributions, and expected value of sums of random variables

Sections 4.8.1, 4.8.2, 4.9 (read 4.10 on own)

F 10/9

Catch-up and review

 

M 10/12

Exam 1 (Chapters 1 – 4)

Reread Chapters 1 – 4

W 10/14

Continuous random variables (definition, expectation, and variance) and uniform distribution

Sections 5.1 – 5.3

F 10/16

Normal distribution (and normal approximation to the binomial distribution)

Sections 5.4

M 10/19

Exponential and Gamma distributions

Sections 5.5, 5.6.1

W 10/21

Distribution of a function of a random variable

Section 5.7

F 10/23

No class – Reading Period

 

M 10/26

Joint distributions and independent random variables

Sections 6.1 – 6.2

W 10/28

Sums of independent random variables  

Section 6.3

F 10/30

Conditional distributions, smallest and largest order statistics

Sections 6.5 – 6.6

M 11/2

Order statistics and review problems

Section 6.6

W 11/4

Joint probability distributions and expectations of sums

Sections 6.7, 7.1 – 7.2

F 11/6

Covariance, variance of sums, and correlations

Section 7.4

M 11/9

Covariance examples and moment generating functions

Sections 7.4, 7.7

W 11/11

Moment generating functions

Section 7.7

F 11/13

Central Limit Theorem

Section 8.3

M 11/16

Catch-up and review

 

F 11/20

Exam 2 (Chapters 5 – 7, 8.3) – 6:30pm

Reread Chapters 5 – 7, 8.3