Math/Stat 431 Introduction to the Theory of Probability

Fall 2013 Lectures 1 and 3, MWF

WAITLISTED STUDENTS: Please understand that emailing instructors to bypass the waitlist is not helpful. I have had to stop responding to such emails. A fourth section may open soon.

Meetings: MWF, Lecture 1 9:55-10:45, Lecture 2 1:20-2:10, Van Vleck B115

Instructor: Timo Seppäläinen

Office: 419 Van Vleck. Office Hours: Mondays and Wednesdays after classes, other times by appointment.

Phone: 263-2812

E-mail: seppalai at math dot wisc dot edu (I can answer questions by email.)

This is the course homepage that also serves as the syllabus for the course. Here you will find homework assignments, our weekly schedule, and updates on scheduling matters. The Mathematics Department also has a general information page on this course.

Course description

Math 431 is an introduction to probability theory, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include methods of counting (combinatorics), axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.

Probability theory is ubiquitous in natural science, social science and engineering, so this course can be valuable in conjunction with many different majors. 431 is not a course in statistics. Statistics is a discipline mainly concerned with analyzing and representing data. Probability theory forms the mathematical foundation of statistics, but the two disciplines are separate.

From a broad intellectual perspective, probability is one of the core areas of mathematics with its own distinct style of reasoning. Among the other core areas are analysis, algebra, geometry/topology, logic and computation.

To go beyond 431 in probability you should take next 521 Analysis, and after that one or both of these: 632 Introduction to Stochastic Processes and 635 Introduction to Brownian Motion and Stochastic Calculus.

Prerequisites

To be technically prepared for Math 431 one needs to be comfortable with the language of sets and calculus, including multivariable calculus, and be ready for abstract reasoning. Probability theory can seem very hard in the beginning, even after success in past math courses.

Textbook

A First Course in Probability, Eighth Edition, by S. Ross. Note that the 8th edition is not the newest edition. But other editions of Ross's book cover the subject matter also. The only possible harm from using a different edition is that you may have to look up the homework problems from the 8th edition.

Evaluation

Course grades will be based on homework and quizzes, three midterm exams, and a comprehensive final exam. Midterm exams will be in the evenings. No calculators, cell phones, or other gadgets will be permitted in exams and quizzes, only pencil and paper.

Lecture 1 final exam 12/19/2013, Thursday 12:25 PM - 2:25 PM.
Lecture 3 final exam 12/18/2013, Wednesday 5:05 PM - 7:05 PM.

Here are grade lines that can be guaranteed in advance. A percentage score in the indicated range guarantees at least the letter grade next to it.
A [100,90), AB [90,87), B [87,76), BC [76,74), C [74,62), D [62,50), F [50,0].

Weekly schedule

Below is the weekly schedule last time I taught this course from Ross's book. The same topics will be covered in Fall 2013, perhaps in slightly altered order.

Week 1. 1.1-1.5, 2.2-2.3
Week 2. 2.3-2.5, 3.2. Quiz 1 on Chapter 1 on Wednesday.
Week 3. 3.3-3.4. Quiz 2 on Sections 3.2-3.3 on Friday.
Week 4. 3.4-3.5. Exam 1 on Chapters 1-3 on Friday, February 11.
Week 5. 4.1-4.6.
Week 6. 4.7-4.10, moment generating function.
Week 7. 5.1-5.5. Quiz 3 on Chapter 4 on Wednesday.
Week 8. 5.5-5.7. Exam 2 on Chapters 4-5 on Wednesday, March 9.
Week 9. 6.1-6.3. -->
Week 10. 6.3-6.5
Week 11. 7.2, 7.4
Week 12. 7.4-7.5 -->
Week 13. 7.5, 7.7. Exam 3 on Chapters 6-7 Friday, April 22.
Week 14. 8.2-8.4
Week 15. Additional topics, interesting examples, review.

Homework

Homework assignments will be posted here.

Larger list of suggested exercises, not for handing in

Chapter 1 Problems: 5, 7, 8, 9, 10, 11, 13, 15, 16, 19, 20, 25, 26. Theoretical Exercises: 5, 9, 10, 11, 12.
Chapter 2 Problems: 1, 3, 6, 9, 13, 18, 27, 41, 45, 47, 50. Theoretical Exercises: 6, 10, 11.
Chapter 3 Problems: 1, 2, 7, 10, 13, 15, 16, 17, 18, 38, 47, 53, 55, 56, 57, 74, 76, 77, 78, 79, 89, 90. Theoretical Exercises: 1, 2, 5, 6, 9.
Chapter 4 Problems: 1, 4, 8, 20, 21, 22, 23, 38, 43, 50, 51, 52, 59, 60, 61, 72, 75, 84. Theoretical Exercises: 2, 3, 4, 5, 6, 7, 9, 11, 14, 25, 27.
Chapter 5 Problems: 1, 3, 5, 7, 10, 13, 18, 19, 27, 28, 33, 37, 39, 40. Theoretical Exercises: 1, 5, 11, 12, 13, 15, 30, 31.
Chapter 6 Problems: 1, 2, 4, 6, 8, 9, 10, 12, 17, 18, 23, 28, 33, 34, 36, 37, 38, 39, 42, 48. Theoretical Exercises: 2, 5, 6, 9, 15, 17.
Chapter 7 Problems: 9, 16, 19, 21, 22, 26, 30, 32, 37, 45, 47, 57, 75, 76. Theoretical Exercises: 9, 10.
Chapter 8 Problems: 1, 4, 5, 6, 7, 11, 14, 15.

Instructions for homework

Observe rules of academic integrity. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. Plagiarism cases will lead to sanctions.
Homework is collected at the beginning of the class period on the due date. No late papers will be accepted. You can bring the homework earlier to the instructor's office or mailbox.
Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Answers should be simplified as much as possible. If the answer is a simple fraction or expression, a decimal answers from a calculator is not necessary. But for some exercises you need a calculator to get the final answer.
Answers to some exercises are in the back of the book, so answers alone carry no credit. It's all in the reasoning you write down.
Put problems in the correct order and staple your pages together.
Do not use paper torn out of a binder.
Be neat. There should not be text crossed out.
Recopy your problems. Do not hand in your rough draft or first attempt.
Papers that are messy, disorganized or unreadable cannot be graded.

The Math Club provides interesting lectures and other math-related events. Everybody is welcome.