Math/Stat 431 Introduction to the Theory of Probability

Summer 2016

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.

I will use the class email list to send out corrections, announcements, please check your wisc.edu email from time to time.

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. It is also important to be familiar with the basic techniques of counting (we will review these along the way). Probability theory can seem very hard in the beginning, even after success in past math courses.

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. We have also placed A First Course in Probability, by Sheldon Ross, on reserve at the math library (floor B2 of Van Vleck) if you want extra reading and extra practice problems.

Piazza

We will be using Piazza for class discussion. The system is catered to getting you help fast and efficiently from classmates. 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 answer questions. If you have any problems or feedback for the developers, email team@piazza.com.

Evaluation

Course grades will be based on homework (16%), quizzes (4%), two midterm exams (2x20%), and a comprehensive final exam (40%).

(Very important!) Please reserve dates of midterms and the Final in your calendar.

• Midterm exam 1: Thursday, June 30 (in class),

• Midterm exam 2: Thursday, July 21 (in class),

• Final exam: Thursday, April 8 (Different venue, 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.

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.

Quizzes

To help prepare for the midterm exams we will have short in-class "surprise" quizzes. It is your responsibility to attend lectures and quizzes.

Homework

Homework assignments will be posted on the Learn@UW site of the course. Homework assignments are due at the beginning of the class.

• No late homework will be accepted. The lowest score will be dropped.

• Homework is collected at the beginning of the class on the due date.

• You can also submit your solution to the grader directly, but it has to arrive by 2:35pm on the due date.

Other 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.

  • 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.

  • Papers that are messy, disorganized or unreadable cannot be graded.

  • I strongly encourage you to type up your solutions (perhaps using Latex).

Weekly schedule

Here is a tentative weekly schedule, to be adjusted as we go. The numbers refer to sections in lecture notes that can be found at the Learn@UW website.

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