Math 531 Probability Theory

Spring 2017

Meetings: TuTh 9:30-10:45 Van Vleck B115

Instructor: Timo Seppäläinen

Office: 425 Van Vleck. Office Hours: after class, or by appointment.
Phone: 263-3624

E-mail: seppalai and then at math dot wisc dot edu

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

Course description

Math 531 is a mathematically rigorous introduction to probability theory at an advanced undergraduate level. Probability theory is the part of mathematics that studies random phenomena. 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.

This course gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and goes over some classical results of probability theory with proofs, such as the DeMoivre-Laplace limit theorem, the study of simple random walk, and applications of generating functions. Math 531 serves both as a stand-alone undergraduate introduction to probability theory and as a sequel to Math/Stat 431 for students who wish to learn the 431 material at a deeper level and tackle some additional topics.

Probability theory is ubiquitous in natural science, social science and engineering, so this course can be valuable in conjunction with many different majors. 531 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.

After 531 the path forward in probability theory goes as follows. At the undergraduate level there are two courses on stochastic processes: 632 Introduction to Stochastic Processes and 635 Introduction to Brownian Motion and Stochastic Calculus. Another alternative is to take 629 Measure Theory as preparation for graduate probability Math/Stat 733-734.

Prerequisites

A proof-based analysis course (such as UW-Madison Math 421, Math 521, or Math 375-376) or consent of the instructor.

Textbook

Grimmett-Stirzaker: Probability and Random Processes, Oxford University Press, 3rd edition.

We cover selected sections mainly from the first 5 chapters of the book and also some parts of later chapters.

Learn@UW

Homework assignments, solutions to homework, and some additional lecture notes will be posted on Learn@UW.

Piazza

Piazza is an online platform for class discussion. Post your math questions on Piazza and answer other students' questions. If you have any problems or feedback for the developers, email team@piazza.com.

Our class Piazza page is at https://piazza.com/wisc/spring2017/math531_001_sp17/home

Evaluation

Course grades will be based on homework and quizzes (15%), three midterm exams (20%+20%+10%, 10% for your lowest midterm exam), and a comprehensive final exam (35%). Midterm exams will be in class on the following dates:

Exam 1 Tuesday February 7
Exam 2 Tuesday March 7
Exam 3 Thursday April 13

Final exam: Monday May 8, 12:25-2:25 PM, Ingraham 120.

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

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.

[100,89) A, [89,87) AB, [87,76) B, [76,74) BC, [74,62) C, [62,50) D, [50,0] F.

Spring 2017 weekly schedule

Here we record the topics covered. Section numbers refer to Grimmett-Stirzaker. [ASV] section numbers refer to 431 notes on Learn@UW by Anderson-Seppäläinen-Valkó.

Week Tuesday Thursday

1 1/17-20 1.3 Kolmogorov's axioms for a probability space. Examples: finitely many fair coin flips, Poisson distribution on nonnegative integers, infinite sequence of fair coin flips. 1.3 Uniformly random point on a disk. Continuity of probability. Repeated rolls of a fair die produce a six eventually. 1.4 Conditional probability. Product rule, law of total probability. Coin example.

2 1/23-27 1.5 Independence. 1.6 is skipped. 1.7 Gambler's ruin. Other examples for reading. 2.1, 2.3 Random variables, discrete and continuous, PMF, PDF and CDF. 2.2 is skipped for now. HW 1 due.

3 1/20-2/3 2.1, 2.3 Random variables, discrete and continuous, PMF, PDF and CDF. [ASV 1.5, 3.1-3.2]
2.5 Random vectors. Quiz 1. Finish Sections 2.1, 2.3-2.5. Prove characterization of CDF. HW 2 due.

4 2/6-10 Exam 1. 3.1, 3.5 The most important discrete distributions, derived from independent repeated trials: Bernoulli, binomial, geometric, Poisson. [ASV 2.4, 4.4]

5 2/13-17 3.1, 3.5 Poisson distribution. [ASV 4.4]
4.1, 4.5 Continuous distributions: uniform, exponential, normal. [ASV 3.1, 3.5, 4.5] 4.5 Normal distribution. [ASV 3.5]
3.2, 4.2 Independence of random variables. [ASV 2.3, 6.3]
Exam 1 bonus quiz.

6 2/20-24 Expectation of random variables.
3.3, 4.3 Special definitions for discrete and continuous random variables. Poisson and uniform examples. [ASV 3.3]
Development of general definition of expectation from separate lecture notes.
HW 3 due. Continue development of general definition of expectation from separate lecture notes.

7 2/27-3/3 Continue development of general definition of expectation from separate lecture notes. Finish development of expectation.
3.3, 4.3 Formulas for E[g(X)]. Variance.
HW 4 due.

8 3/6-10 Exam 2. 3.4 Computing expectations with indicator random variables. [ASV 8.1]
Examples and properties of variance.

9 3/13-17 Independence and expectation. [ASV 8.2]
3.6, 4.5 Covariance, its properties. Variance of a sum. Examples: indicators, uniform random points in higher dimensions. [ASV 6.1, 6.2, 8.4] 3.6, 4.5 Multinomial distribution. Correlation coefficient. Cauchy-Schwarz inequality. [ASV 6.1, 8.4]
HW 5 due.

3/20-24 SPRING BREAK SPRING BREAK

10 3/27-31 3.6, 4.5 Completion of correlation topic. [ASV 8.4]
3.8, 4.8 Sums of independent random variables. [ASV 7.1, 8.3]
Gamma function and gamma distribution. [ASV p. 153, GS p. 96] Moment generating function: identifying the distribution of a sum, calculating moments. [ASV 5.1, 7.1, 8.3]

11 4/3-7 Separate lecture notes: Markov and Chebyshev inequalities, weak law of large numbers. Convergence in probability and almost surely. Part of this material is in ASV 9.1-9.2. GS 7.2-7.4 cover more material. Separate lecture notes: review of limsup and liminf, convergence of random variables as an event, first Borel-Cantelli lemma.
HW 6 due.

12 4/10-14 Separate lecture notes: strong law of large numbers. Exam 3.

13 4/17-21 Separate lecture notes: central limit theorem for i.i.d. Bernoulli random variables. Confidence intervals. 3.7, 4.6 Conditional distributions: conditional probability mass function, conditional density function, conditional expectation. [ASV 10]

14 4/24-28 3.9, 3.10 Random walk. (Also separate lecture notes.) Further properties of random walk.
4.12 Coupling and Poisson approximation.

15 5/1-4 General central limit theorem for IID sequences with finite variance. Sketch of proof assuming finite MGF. Comparison of normal and Poisson approximation of the binomial. Review of joint densities. Review of convolutions. Review of types of convergence. Application of Borel-Cantelli to prove that convergence in probability implies almost sure convergence along a subsequence

Week	Tuesday	Thursday
1 1/17-20	1.3 Kolmogorov's axioms for a probability space. Examples: finitely many fair coin flips, Poisson distribution on nonnegative integers, infinite sequence of fair coin flips.	1.3 Uniformly random point on a disk. Continuity of probability. Repeated rolls of a fair die produce a six eventually. 1.4 Conditional probability. Product rule, law of total probability. Coin example.
2 1/23-27	1.5 Independence. 1.6 is skipped.	1.7 Gambler's ruin. Other examples for reading. 2.1, 2.3 Random variables, discrete and continuous, PMF, PDF and CDF. 2.2 is skipped for now. HW 1 due.
3 1/20-2/3	2.1, 2.3 Random variables, discrete and continuous, PMF, PDF and CDF. [ASV 1.5, 3.1-3.2] 2.5 Random vectors. Quiz 1.	Finish Sections 2.1, 2.3-2.5. Prove characterization of CDF. HW 2 due.
4 2/6-10	Exam 1.	3.1, 3.5 The most important discrete distributions, derived from independent repeated trials: Bernoulli, binomial, geometric, Poisson. [ASV 2.4, 4.4]
5 2/13-17	3.1, 3.5 Poisson distribution. [ASV 4.4] 4.1, 4.5 Continuous distributions: uniform, exponential, normal. [ASV 3.1, 3.5, 4.5]	4.5 Normal distribution. [ASV 3.5] 3.2, 4.2 Independence of random variables. [ASV 2.3, 6.3] Exam 1 bonus quiz.
6 2/20-24	Expectation of random variables. 3.3, 4.3 Special definitions for discrete and continuous random variables. Poisson and uniform examples. [ASV 3.3] Development of general definition of expectation from separate lecture notes. HW 3 due.	Continue development of general definition of expectation from separate lecture notes.
7 2/27-3/3	Continue development of general definition of expectation from separate lecture notes.	Finish development of expectation. 3.3, 4.3 Formulas for E[g(X)]. Variance. HW 4 due.
8 3/6-10	Exam 2.	3.4 Computing expectations with indicator random variables. [ASV 8.1] Examples and properties of variance.
9 3/13-17	Independence and expectation. [ASV 8.2] 3.6, 4.5 Covariance, its properties. Variance of a sum. Examples: indicators, uniform random points in higher dimensions. [ASV 6.1, 6.2, 8.4]	3.6, 4.5 Multinomial distribution. Correlation coefficient. Cauchy-Schwarz inequality. [ASV 6.1, 8.4] HW 5 due.
3/20-24	SPRING BREAK	SPRING BREAK
10 3/27-31	3.6, 4.5 Completion of correlation topic. [ASV 8.4] 3.8, 4.8 Sums of independent random variables. [ASV 7.1, 8.3] Gamma function and gamma distribution. [ASV p. 153, GS p. 96]	Moment generating function: identifying the distribution of a sum, calculating moments. [ASV 5.1, 7.1, 8.3]
11 4/3-7	Separate lecture notes: Markov and Chebyshev inequalities, weak law of large numbers. Convergence in probability and almost surely. Part of this material is in ASV 9.1-9.2. GS 7.2-7.4 cover more material.	Separate lecture notes: review of limsup and liminf, convergence of random variables as an event, first Borel-Cantelli lemma. HW 6 due.
12 4/10-14	Separate lecture notes: strong law of large numbers.	Exam 3.
13 4/17-21	Separate lecture notes: central limit theorem for i.i.d. Bernoulli random variables. Confidence intervals.	3.7, 4.6 Conditional distributions: conditional probability mass function, conditional density function, conditional expectation. [ASV 10]
14 4/24-28	3.9, 3.10 Random walk. (Also separate lecture notes.)	Further properties of random walk. 4.12 Coupling and Poisson approximation.
15 5/1-4	General central limit theorem for IID sequences with finite variance. Sketch of proof assuming finite MGF. Comparison of normal and Poisson approximation of the binomial. Review of joint densities.	Review of convolutions. Review of types of convergence. Application of Borel-Cantelli to prove that convergence in probability implies almost sure convergence along a subsequence

Instructions for homework

Homework is collected in class on the due date, or alternately can be brought to the instructor's office or mailbox by 2 PM on the due date. No late papers will be accepted. You can bring the homework earlier to the instructor's office or mailbox.
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. You are certainly encouraged to discuss the problems with your fellow students, but in the end you must write up and hand in your own solutions.
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.
As always in mathematics, numerical 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.