Math 401: Probability

Math 401: Probability

Michael W. Trosset

The following information is for Fall 2002:

General Description
This course is devoted to mathematical probability. Topics include the Kolmogorov probability axioms, conditioning and independence, random variables, various discrete and continuous probability distributions, expectation and various limit theorems. Most assignments involve solving problems and/or deriving elementary propositions.

Roughly speaking, introductory probability courses come in three types, classified by the mathematical background that they assume:

Finite Math. At this level (usually high school or lower division undergraduate), most treatments of probability restrict attention to experiments that involve finite sample spaces of equally likely outcomes, e.g. tossing a fair coin, rolling a fair die, or dealing a hand of cards. Counting arguments (combinatorics) are strongly emphasized. Math 401 introduces a few elementary counting arguments, but mainly emphasizes other topics. Students interested in combinatorics should consider taking Math 432.

Calculus. At this level (usually upper division undergraduate), most treatments of probability distinguish between discrete and continuous random variables. The former are treated rigorously, the latter somewhat informally. Important families of distributions (binomial, normal, etc.) are studied. Formal manipulations involving joint, marginal, and conditional distributions are crucial and require the student to have mastered the basic skills of multivariable calculus. Fundamental results such as the Weak Law of Large Numbers and the Central Limit Theorem are introduced. Math 401 is such a treatment, although it occasionally attempts to offer brief glimpses of probability from a more advanced, measure-theoretic perspective.

Measure Theory. At this level (usually graduate), most treatments of probability are purely theoretical. The subject is unified by relying on such tools as Lebesgue integration with respect to a general measure and the Radon-Nikodym Theorem. Familiarity with such tools is either assumed or developed as part of the course. Math 401 does neither, although it does occasionally attempt to offer brief glimpses of the elegance of this perspective.

Prerequisites
Required: Math 211, 212, 214.
Recommended: Math 311.

Basic Information
Math 401 will meet on Monday-Wednesday-Friday, from 1:00 to 1:50 p.m., in Room 240 of Small Hall. The final exam is scheduled for 1:30 p.m. on Tuesday, December 17.

Tentative Office Hours
I will delighted to meet with you in my office, Room 127 of Jones Hall. I usually will be available after class, from 2:00 to 3:30 p.m. Alternatively, please contact me (before/after class, send email to trosset@math.wm.edu, telephone 1-2040) to schedule an appointment at a mutually convenient time..

Attendance
Class attendance is not formally required, but it is strongly encouraged. Ignorance of supplementary material presented---or announcements made---by the instructor due to absence from class is never excusable. In class you are expected to behave appropriately, e.g. please refrain from conversing with other students while the instructor is lecturing.

Text
INTRODUCTION TO PROBABILITY THEORY, 1971, by P.G. Hoel, S.C. Port, and C.J. Stone. This book is available at the William & Mary bookstore.

Syllabus
The following topics approximate my lectures in Fall 2001, when Math 401 met on Tuesdays and Thursdays. I expect to cover the same material in Fall 2002, but with more lectures of shorter duration.

Lecture 1. Probability Spaces. (Section 1.2)
Lecture 2. Properties of Probability. (Section 1.3)
Lecture 3. Conditional Probability and Independence. (Sections 1.4 and 1.5)
Lecture 4. Elementary Combinatorics. (Sections 2.1 through 2.5)
Lecture 5. Discrete Random Variables. (Sections 3.1 and 3.2)
Lecture 6. Discrete Random Vectors and Independence. (Sections 3.3 and 3.4)
Lecture 7. Bernoulli Trials and Sums of Independent Random Variables. (Sections 3.5 and 3.6)
Lecture 8. Expectation and Moments. (Sections 4.1 through 4.3)
Lecture 9. Covariance and Correlation. (Sections 4.5 and 4.6)
Lecture 10. The Weak Law of Large Numbers. (Section 4.6)
Lecture 11. Continuous CDFs and PDFs. (Sections 5.1 and 5.2)
Lecture 12. Change of Variables, Symmetry. (Sections 5.2.1 and 5.2.2)
Lecture 13. Normal Distributions, Exponential Distributions. (Section 5.3)
Lecture 14. Gamma Distributions, Inverse Distributions. (Sections 5.3 and 5.4)
Lecture 15. Bivariate Distributions. (Section 6.1)
Lecture 16. Sums of Random Variables. (Section 6.2.1)
Lecture 17. Conditional PDFs, Multivariate Distributions (Sections 6.3 and 6.4).
Lecture 18. Sampling Distributions, Change of Variables. (Sections 6.6 and 6.7)
Lecture 19. Expectation. (Sections 7.1 and 7.2)
Lecture 20. Moments, Conditional Expectation. (Sections 7.3 and 7.4)
Lecture 21. The Central Limit Theorem. (Section 7.5)
Lecture 22. Complex Numbers and Characteristic Functions. (Section 8.2)
Lecture 23. The Inversion and Continuity Theorems. (Section 8.3)
Lecture 24. Proof of WLLN. (Section 8.4)
Lecture 25. Proof of CLT. (Section 8.4)

Grades
For each student, a weighted course average will be calculated as follows:

Performance on weekly homework assignments (30%). Unless stated otherwise, homework will be due the Friday following the week in which it is assigned. Please staple multiple pages together and trim ragged edges! I will accept any reasonable excuse for late homework, provided it is offered in advance of the due date. Collaboration on homework with other students is both permitted and encouraged.
Performance on midterm tests (40%). Tests will be take-home, open book and open notes. Test 1 will cover the material in Chapters 1-4. Test 2 will cover the material in Lectures 11-20 Chapters 5-6. Collaboration on tests with other students is not permitted under any circumstances and will be considered a violation of the William & Mary honor code.
Performance on final exam (30%). The final will be comprehensive (Chapters 1-8). It will be in-class, closed book and closed notes. You will be permitted the use of a calculator and three formula sheets, i.e. three 8.5 x 11 inch sheets of paper on which you may record any information you please. Collaboration on the final exam with other students is not permitted under any circumstances and will be considered a violation of the William & Mary honor code.

I will assign semester grades on the basis of course averages, attempting to identify clusters of similarly performing students.

Homework/Test Solutions
Here are links to PDF files that contain homework solutions: Homework 1, Homework 2, Homework 3, Homework 4, Homework 5, Homework 6, Homework 7, Homework 8, Homework 10.