# Math/Stat 431 Introduction to the Theory of Probability

### Fall 2016 Lecture

 Meetings: TuTh 9:30AM-10:45AM Van Vleck B123 Instructor: Benedek Valkó Office: 409 Van Vleck. Office Hours: W 3:30PM-4:30PM or by appointment E-mail: valko at math dot wisc dot edu

### Course description

Math 431 is an introduction to the theory of probability, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include 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.

Math 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 Math 521 - Analysis, and after that one or both of these: Math 632 - Introduction to Stochastic Processes and Math 635 - Introduction to Brownian Motion and Stochastic Calculus. Those who would like a proof based introduction to probability could consider taking  Math 531 - Probability Theory (531 requires a proof based course as a prerequisite).

### Where is probability used?

Probability theory is ubiquitous in natural science, social science and engineering, so this course can be valuable in conjunction with many different majors. Aside from being a beautiful subject in and of itself, is used throughout the sciences and industry.  For example, in biology many models of cellular phenomena are now modeled probabilistically as opposed to deterministically. As for industry, many models used by insurance and financial companies are probabilistic in nature. Thus, those wishing to go into actuarial science or finance need to have a solid understanding of probability. Probabilistic models show up in the study of networks, making probability theory useful for those interested in computer science and information technology.

### 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. Basic techniques of counting is also useful, but we will review these along the way. Probability theory can seem very hard in the beginning, even after success in past math courses.

### Learn@UW

We will use the Learn@UW website of the course to post homework assignments and solutions. The lecture notes will also be posted there.

### Textbook

The course follows lecture notes by David Anderson, Timo Seppäläinen, and Benedek Valkó.  These will be provided to the students at no cost.  The textbook can be found at the Learn@UW website.  The following textbooks can be used as a resource for extra practice problems. They have been placed on reserve at the math library (floor B2 of Van Vleck).
• A first course in Probability, by Sheldon Ross
• Probability, by Jim Pitman.
• Probability, statistics, and stochastic processes, by Peter Olofsson.
• Probability and random processes, by Geoffrey Grimmett and David Stirzaker.

Piazza
We will be using Piazza for class discussion.  The system is catered to getting you help fast and efficiently from classmates and myself.  Rather than emailing math questions to me, I encourage you to post your questions on Piazza. The students (and instructors) from the other three sections of 431 will have access to the same page, and can ask and answer questions. If you have any problems or feedback for the developers, email team@piazza.com.

Find our class Piazza page at: https://piazza.com/wisc/fall2016/math431_fa16/home

Evaluation
Course grades will be based on homework and  quizzes (15%), two midterm exams (2x25%), and a comprehensive final exam (35%).
Midterm exams will be in the evenings of the following dates.
• Midterm exam 1: Thursday, October 13, 7:15-8:45 PM, 1310 Sterling
• Midterm exam 2: Wednesday, November 30, 7:15-8:45 PM, 1310 Sterling
• Final exam: Tuesday, December 20, 12:25PM - 2:25PM, Room TBA
No calculators, cell phones, or other gadgets will be permitted in exams and quizzes, only pencil and paper.
The final grades will be determined according to the following scale:
A: [100,89), AB: [89,87), B: [87,76), BC: [76,74), C: [74,62), D: [62,50), F: [50,0].
There will be no curving in the class, but the instructor reserves the right to modify the final grade lines.

### Quizzes

To help prepare for the midterm exams we will have short in-class quizzes during the first couple of weeks.
• 1st quiz: September 13, Tuesday, end of class.
• 2nd quiz: September 20, Tuesday, end of class.
• 3rd quiz: September 27, Tuesday, end of the class
• 4th quiz: October 4, Tuesday, end of the class

There will be no more quizzes in the semester.

### Homework

Homework assignments will be posted on the Learn@UW site of the course. Weekly homework assignments are usually due on Thursday at the beginning of the class. You can also submit your solution in an electronic form via Learn@UW, but it has to arrive by 9:30AM on the due date. Note that there is a (short) homework assignment due on the first Thursday (September 8).
Some homework assignments will contain bonus problems for those who would like extra challenge. The points from the bonus problems will be converted into extra credit at the end of the semester.

### 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 assignments will be accepted. You can bring the homework earlier to the instructor's office.
• Working in groups on homework assignments is strongly encouraged; however, every student must write their own assignments.
• 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. For some exercises you will 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.