Math/Stat 431 Introduction to the Theory of Probability

Spring 2018 Lecture 001

I will use the class email list and CANVAS 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.

Learning Outcomes

Students will be able to state, explain, and apply the axioms, principal results, definitions, and theorems of undergraduate probability theory. This includes concepts such as probability spaces, random variables and random vectors and their probability distributions, conditional probability and expectation, independence, law of large numbers, and the central limit theorem. Students will be able to calculate probabilities and expectations in simple model problems. Students will be able to use probability theory to model simplified real-world situations with random outcomes. In particular, students can identify and apply the most common probability distributions, including the Bernoulli, binomial, geometric, Poisson, uniform, normal, and exponential distributions. Students will be able to estimate probabilities with the normal and Poisson approximations and the Markov and Chebyshev inequalities.

Textbook

Introduction to Probability by David Anderson, Timo Seppäläinen, and Benedek Valkó.

Evaluation

Course grades will be based on homework (16%), quizzes (4%), two midterm exams (2x20%), and a comprehensive final exam (40%). Midterm exams will be in the evenings of these dates. (Very important!) Please reserve these slots in your calendar.

• Midterm 1: February 28 (Wed) 7:15-8:45 PM Van Vleck B130

• Midterm 2: April 18 (Wed), 7:15-8:45 PM Van Vleck B130

• Final exam: May 7 (Mon), 10:05AM - 12:05

• No calculators, cell phones, or other gadgets will be permitted in exams and quizzes.

• 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.

• No make-up midterm exams will be considered less there is a conflict with another university-related event (in which case case you need to contact me as soon as possible), or a last-minute medical/family emergency (you need to show proof of such emergency, e.g., doctor's notes).

• Here are the old exams from the Library: https://uwmadison.app.box.com/v/math431

Quizzes

To help prepare for the exams we will have short in-class "surprise" quizzes (on Wednesdays). It is your responsibility to attend lectures and quizzes. The lowest score will be dropped.

Homework

Homework assignments will be posted on the CANVAS. Weekly homework assignments are due Fridays at the beginning of the class.

(Homework 1 will be due Friday Feb 2.)

• 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 9:55am on the due date.

• Our grader is Yuqing Dai (ydai39@wisc.edu)

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).

Learn@UW / CANVAS

We will use the CANVAS to post homework assignments and solutions. You should have access to it through Learn@UW.

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.

Tentative schedule

Here is a tentative weekly schedule, to be adjusted as we go. The numbers refer to sections in the textbook.

Getting extra help:

• • Mathlab: Free, drop-in tutorial program offered by math department. Staffed primarily by Mathematics Department teaching assistants.

    • Location: B227/B205 Van Vleck Hall (across from the Mathematics Library)

    • Hours: Monday through Thursday, 3:30 - 8:30 pm, and Sunday 3:30 - 6:50 pm.

    • A math tutor for 431, Liding Yao, will be in B205 during Wed 14:25-15:55, Fri 14:25-17:25, Sun 12:30-15:00.

ACADEMIC INTEGRITY

By enrolling in this course, each student assumes the responsibilities of an active participant in UW-Madison's community of scholars in which everyone's academic work and behavior are held to the highest academic integrity standards. Academic misconduct compromises the integrity of the university. Cheating, fabrication, plagiarism, unauthorized collaboration, and helping others commit these acts are examples of academic misconduct, which can result in disciplinary action. This includes but is not limited to failure on the assignment/course, disciplinary probation, or suspension. Substantial or repeated cases of misconduct will be forwarded to the Office of Student Conduct & Community Standards for additional review. For more information, refer to https://conduct.students.wisc.edu/academic-integrity/.

ACCOMMODATIONS FOR STUDENTS WITH DISABILITIES

McBurney Disability Resource Center syllabus statement: “The University of Wisconsin-Madison supports the right of all enrolled students to a full and equal educational opportunity. The Americans with Disabilities Act (ADA), Wisconsin State Statute (36.12), and UW-Madison policy (Faculty Document 1071) require that students with disabilities be reasonably accommodated in instruction and campus life. Reasonable accommodations for students with disabilities is a shared faculty and student responsibility. Students are expected to inform faculty [me] of their need for instructional accommodations by the end of the third week of the semester, or as soon as possible after a disability has been incurred or recognized. Faculty [I], will work either directly with the student [you] or in coordination with the McBurney Center to identify and provide reasonable instructional accommodations. Disability information, including instructional accommodations as part of a student's educational record, is confidential and protected under FERPA.” http://mcburney.wisc.edu/facstaffother/faculty/syllabus.php

DIVERSITY & INCLUSION

Institutional statement on diversity: “Diversity is a source of strength, creativity, and innovation for UW-Madison. We value the contributions of each person and respect the profound ways their identity, culture, background, experience, status, abilities, and opinion enrich the university community. We commit ourselves to the pursuit of excellence in teaching, research, outreach, and diversity as inextricably linked goals.

The University of Wisconsin-Madison fulfills its public mission by creating a welcoming and inclusive community for people from every background – people who as students, faculty, and staff serve Wisconsin and the world.” https://diversity.wisc.edu/

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