# Math/Stat 431 Introduction to the Theory of Probability

### Fall 2016 Lecture 002 MWF

 Meetings: MWF 8:50-9:40 Van Vleck B119 Instructor: Timo Seppäläinen Office: 425 Van Vleck. Office Hours: Mondays and Wednesdays 10-11, other times 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 431 is an introduction to probability theory, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include methods of counting (combinatorics), 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.

Probability theory is ubiquitous in natural science, social science and engineering, so this course can be valuable in conjunction with many different majors. 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 521 Analysis for basic mathematical groundwork. After that you have several options for subsequent undergraduate probability courses. 531 Probability Theory is a proof-based introduction to probability that covers the material of 431 in a deeper way and tackles some additional topics. There are two courses on stochastic processes: 632 Introduction to Stochastic Processes and 635 Introduction to Brownian Motion and Stochastic Calculus.

### 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. Probability theory can seem very hard in the beginning, even after success in past math courses.

### Textbook

The course follows lecture notes authored by David Anderson, Timo Seppäläinen, and Benedek Valk�. These will be provided to the students at no cost.

### Learn@UW

We use Learn@UW to post homework assignments, solutions to homework, and lecture notes.

### Library reserves

The following books are on reserve in the library. Going over different presentations of the material can be helpful. Especially the first two are good sources for additional practice problems.
• 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.
You can find old 431 exams in the Math Library Course Reserves. But be aware that old exams are not necessarily a perfect fit for this semester's course.

### Piazza

We use Piazza for class discussion. Post your math questions on Piazza. Students and instructors from all sections of 431 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.

Our class Piazza page is 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 are in the evenings of the following dates.
• Midterm exam 1: Thursday, October 13, 7:15-8:45 PM. Sterling Hall 1310.
• Make-up for midterm exam 1: Friday, October 14, 7:15-8:45 AM. Van Vleck B115.
• Midterm exam 2: Wednesday, November 30, 7:15-8:45 PM.
• Final exam: Tuesday 12/20/2016, 12:25-2:25 PM, Soc Sci 6102.
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.

### Quizzes

There will be quizzes at the end of Wednesday classes during weeks 2-5, to get used to problem solving under time pressure. We can have quizzes later on in the semester too in order to have some testing in the long period between the two midterm exams.

### Instructions for homework

• Homework is collected at the beginning of the class period 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.
• 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.
• 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.

### Fall 2016 daily schedule

Here we record the sections of the notes as we go through them.

Week Monday Wednesday Friday
1 9/6-9/9 1.2 Axioms of probability. 1.3 Sampling.
2 9/12-9/16 1.3 Sampling. 1.4 Infinitely many outcomes. Review of geometric series. 1.5 Using inclusion-exclusion and complements to calculate probabilities. Quiz. 1.6 Random variables. Probability mass function of a discrete random variable.
3 9/19-9/23 1.6 Probability distribution of a random variable. 2.1 Conditional probability. 2.1 Conditional probability. 2.2 Bayes formula. Quiz. 2.2 Bayes formula. 2.3 Independence.
4 9/26-9/30 2.3 Independence of 3 or more events. Independence of random variables. 2.4 Independent trials, Bernoulli and binomial distribution. Quiz. 2.4 Independent trials, geometric distribution. 2.5 Conditional independence.
5 10/3-7 3.1 Probability mass functions, probability density functions. 3.2 Expectation EX for discrete and continuous random variables. Quiz. 3.2 Expectation E[g(X)] and variance Var(X) for discrete and continuous random variables.
6 10/10-14 Review of independent trials. 3.2 Expectation and variance of aX+b. 3.3 Normal distribution. Review for Exam 1.
Exam 1 on Thursday night.
3.3 Normal distribution.
7 10/17-21 4.1 Central limit theorem and the normal approximation of the binomial. 4.1 Normal approximation of the binomial with continuity correction. Law of large numbers for binomial. Estimating an unknown success probability. 4.1 Confidence intervals.
8 10/24-28 4.2 Poisson distribution and Poisson approximation of the binomial. Comparison of Poisson and normal approximation of the binomial. Quiz on Section 4.1. 4.3 Exponential distribution. 5.1 Moment generating function, exponential example.
9 10/31-11/4 5.1 Moment generating function, Poisson example, finding moments, identifying distributions. 5.2 Distribution of a function of a random variable. 6.1 Joint distribution of discrete random variables. Multinomial distribution.
10 11/7-11 6.1 Independence in terms of probability mass functions. 6.2 Jointly continuous random variables. 6.2 Jointly continuous random variables. Quiz on Chapter 5. 6.2 Jointly continuous random variables and their independence.
11 11/14-18 7.1 Sums of independent random variables. 7.2 Excheangable random variables. 7.2 Excheangable random variables. 8.1 Linearity of expectation, indicator method.
12 11/21-23 8.2 Independence and expectations. 8.2 Coupon collector's problem. 8.3 Moment generating function of a sum. Thanksgiving break.

### Looking ahead: tentative weekly schedule for fall 2016

Here is a tentative weekly schedule that looks into the future, to be adjusted as we go. Section numbers refer to the lecture notes.

• Week 1. (9/6-9) Axioms of probability, sampling, review of counting, infinitely many outcomes, review of the geometric series (Sections 1.1-1.3).
• Week 2. (9/12-16) Rules of probability, random variables (Sections 1.4-1.5).
• Week 3. (9/19-23) Conditional probability, Bayes formula, independence, independent trials (Sections 2.1-2.4).
• Week 4. (9/26-30) Independent trials, birthday problem, probability distribution of a random variable (Sections 2.4-2.5, 3.1).
• Week 5. (10/3-7) Expectation and variance, Gaussian distribution (Sections 3.2-3.3).
• Week 6. (10/10-14) Normal approximation and law of large numbers for the binomial distribution (Sections 4.1). Midterm Exam 1.
• Week 7. (10/17-21) Normal approximation, Poisson approximation, Exponential distribution (Sections 4.1-4.3)
• Week 8. (10/24-28) Moment generating function, distribution of a function of a random variable (Sections 5.1-5.2).
• Week 9. (10/31-11/4) Joint distributions (Sections 6.1-6.3).
• Week 10. (11/7-11) Sums of independent random variables, exchangeability (Sections 7.1-7.2).
• Week 11. (11/14-18) Expectations of sums and products, coupon collector's problem (Sections 8.1-8.3).
• Week 12. (11/21-23) Covariance and correlation (Sections 8.4). Thanksgiving break.
• Week 13. (11/28-12/2) Markov's and Chebyshev's inequalities, law of large numbers, central limit theorem (Sections 9.1-9.3). Midterm Exam 2.
• Week 14. (12/5-9) Conditional distributions (Sections 10.1-10.3).
• Week 15. (12/12-15) Conditional distributions, review (Sections 10.1-10.3).

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