## General Information

### Description

This course is meant to provide an introduction to Probability Theory for student in mathematics. The focus will be on the basic ideas of probability and on the fundamental mathematical results of the theory as well as their application in real world situations. Thus, this class will contain formal definitions and rigorous proofs together with applications.

### Prerequisites:

MATH 2551 or MATH 2X51 or MATH 2561 or MATH 2401 or MATH 24X1 or MATH 2411 or MATH 2605 or MATH 2550) AND (MATH 2106 or CS 2051 or MATH 3012

### Course Goals and Learning Outcomes;

My goal is to teach you to basics notion of Probability Theory. This starts with the notion of a probability model and how to build such a model to describe a real situation. This will include a overview of the most important examples such as binomial, Poisson, exponential and normal models. After this I will discuss the basic conceptual tool to analyze this models, such as conditioning and independence. Finally I will discuss the most important results of basic probability: the law of large numbers and the central limit theorem. Time permitting I will introduce the simplest examples of stochastic processes: Markov chains and random walks together with their basics properties. At the end of the class, you will be able to develop and analyze rigorously a probabilistic argument.

There will be two midterms and one final. The midterms and final will be in a take home format. I will post a set of exercises meant to use the material learnt in class. You will have a couple of days to work out the exercises. During this period I'll be available to answer questions. The date of the midterms are tentatively set for Tuesday 9/27 and Tuesday 11/01. .
I will also assign homework from the textbook. This are mostly meant to check that you are following along with the class material.
The final grade will be based on HWs (10%), midterms (50%) and final (40%).
We will also discuss personal projects to be completed together with the final. To prepare a good project it is necessary to start as soon as possible. I can propose subjects for possible projects but I'd prefer if you find a problem involving probability theory you are interested in and that can be analyzed with the tools you will learn in class.

## Course Materials

### Textbook

The class text book is:

G. Grimmett and D. Welsh, Probability: An Introduction. Oxford, 2nd Edition

### Web page

The weekly evolution of the class will be posted on the class web page, together with HW and extra class material. You can also check the we pages of previous editions of this class: Spring 2019 and Spring 2021

## Topic Outline:

• Basics: The probability framework, Conditional probability, Bayes’ theorem, Independent events.
• Random Variables: Discrete and Continuous r.v., Joint distributions, Expectations, variance, covariance
• Convergence Theorems: Convergence in probability and in distribution, Law of Large Numbers, Central Limit Theorem, Large deviations.
• Optional arguments (time permitting): Markov Chains, Random Walks, Convergence to Equilibrium.

## Class development:

First week.

Material covered: Chapter 1 sections 1.1 to 1.5.
Lecture notes: 8/25

Exercises: 1.10, 1.17, 1.21, 1.27, 1.30.

Second week.

Material covered: Chapter 1 sections 1.6 to 1.9.
Lecture notes: 8/30, 9/1

Exercises: 1.44, 1.52.
Problems: 9, 14, 17.

First HW collection: 9/1. HW must be uploaded to canvas before the beginning of class.

Solution set for the first HW.

Third week.

Chapter 2 sections 2.1, 2.2, 2.4.
Lecture notes: 9/6, 9/8.

Exercises: 2.10, 2.11, 2.24.

Forth week.

Material covered: Chapter 2 sections 2.3 and 2.5, Chapter 3 sections 3.1 and 3.2.
Lecture notes: 9/15, 9/19.
Class recording: 9/15, 9/19.

Exercises: 3.8.
Problems for Chapter 2: 4, 5, 7.

Second HW collection: 9/20. Solution set for the second HW.

Fifth week.

Material covered: Chapter 3 section 3.3 to 3.5 and Chapter 4 section 4.1 and 4.2.
Lecture notes: 9/20, 9/22.
Class recording: 9/20.

Exercises: 3.25, 3.42, 4.18.
Problems for Chapter 3: 4, 7.

Sixth week.

Material covered: Chapter 4 section 4.3 and 4.4 and Chapter 5 section 5.1.
Lecture notes: 9/27, 9/29 review.

Exercises: 4.41, 5.13.
Problems for Chapter 4: 5.

The first midterm will be posted on Tuesday 10/4 and will be due on Thursday 10/6 before class. It will cover Chapter 1 to 4 of the book. Here is some practice material from previous years: Spring 19, Spring 20, Spring 21 (take home). Some more examples thanks to prof. Houdré: Spring 18 and Fall 18.

Third HW collection: 10/4. Solution set for the third HW.

Solution set for the first midterm.

Seventh week.

Material covered: Chapter 5 sections 5.2 and 5.6
Lecture notes: 10/04, 10/06.

Exercises: 5.18, 5.30, 5.32, 5.54.
Problems for Chapter 5: 7, 11

Eighth week.

Material covered: Chapter 6 sections 6.1 to 6.4.
Lecture notes: 10/13.

Exercises: 6.14, 6.26, 6.36, 6.45.

Fourth HW collection: 10/25.

Ninth week.

Material covered: Chapter 6 sections 6.5 to 6.8.
Lecture notes: 10/20

Exercises: 6.55, 6.61, 6.70, 6.80.
Problems: 6, 20

Tenth week.

Material covered: Chapter 7 section 2 to 4.
Lecture notes: 10/25, 10/27

Exercises: 7.10, 7.36.

Solution set for the forth HW.

The second midterm will be on Tuesday 11/8 and will be due on Thursday 11/10 before class. It will cover up to Chapter 6 included and concentrate on Chapter 5 and 6 of the book. Here is some practice material from previous years: Spring 19, Spring 21. Some more examples thanks to prof. Houdré: Spring 18 and Fall 18.

Eleventh week.

Material covered: Chapter 7 sections 5 and 6.
Lecture notes: 11/1, 11/3

Exercises: 7.60, 7.75, 7.96.
Problems for chapter 7: 8, 11.

Fifth HW collection: 11/22.

If you want to work on a personal project for improving your final grade this is the moment to decide. Here you can find a couple of past project I enjoyed. The project should be 5 to 10 page long and contain a complete discussion of a probability related problem. Please see me after class or at HO for more information.

Solution set for the second midterm.

Twelfth week.

Material covered: Chapter 8 sections 1 to 3.
Lecture notes: 11/8, 11/10

Exercises: 8.10, 8.11, 8.21

Thirteenth week.

Material covered: Chapter 8 sections 4 to 5.
Lecture notes: 11/15, 11/17

Exercises:8.32, 8.42

Problems for chapter 8: 5, 14

Fourteenth week.

Material covered: Introduction to Stochastic Processes.
Lecture notes: 11/22

Fifteenth week.

Material covered: Markov Chains.
Lecture notes: 11/29, 12/01

The final will be posted on Tuesday 12/13 around 11am and will be due on Thursday 12/15 before 2:10pm. It will cover from Chapter 1 to Chapter 8 included. Here is some practice material from previous years: Spring 19, Spring 21.