Meetings: TuTh 9:3010:45 Van Vleck B115 
Instructor: Timo Seppäläinen 
Office: 425 Van Vleck. Office Hours: after class, or by appointment. 
Phone: 2633624 
Email: 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.
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 DeMoivreLaplace limit theorem, the study of simple random walk, and applications of generating functions. Math 531 serves both as a standalone 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 733734.
We cover selected sections mainly from the first 5 chapters of the book and also some parts of later chapters.
Our class Piazza page is at https://piazza.com/wisc/spring2017/math531_001_sp17/home
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:
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.
Week  Tuesday  Thursday 

1  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 
1.5 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 
2.1, 2.3 Random variables, discrete and continuous, PMF, PDF and CDF. [ASV 1.5, 3.13.2] 2.5 Random vectors. Quiz 1.  Finish Sections 2.1, 2.32.5. Prove characterization of CDF. HW 2 due. 
4 
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 
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  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 
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 
Exam 2.  3.4 Computing expectations with indicator random variables. [ASV 8.1] Examples and properties of variance. 
9 
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. CauchySchwarz inequality. [ASV 6.1, 8.4] HW 5 due. 

SPRING BREAK  SPRING BREAK 
10 
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 
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 9.19.2. GS 7.27.4 cover more material.  Separate lecture notes: review of limsup and liminf, convergence of random variables as an event, first BorelCantelli lemma. HW 6 due. 
12 
Separate lecture notes: strong law of large numbers.  Exam 3. 
13 
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 
3.9, 3.10 Random walk. (Also separate lecture notes.)  Further properties of random walk. 4.12 Coupling and Poisson approximation. 
15 
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 BorelCantelli to prove that convergence in probability implies almost sure convergence along a subsequence 