MATH 3235 Probability Theory
TR 14:00-15:15, Remote synchronous
Office Hours: TR 17:00-18:00, Remote synchronous.
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.
Course Modality: The course will be remote synchronous. Class will be held on Bluejeans using my tablet as a white board. Class notes will be posted on Canvas after each class. I welcome comments and suggestions on how to improve your experience in the class.
Course Requirements and Grading
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 Thursday February 25 and Thursday April 1. .
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 differential equation you are interested in and that can be analyzed with the tools you will learn in class.
The class text book is:
G. Grimmett and D. Welsh, Probability: An Introduction.
Oxford, 2nd Edition
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 2020 and Spring 2021
- 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.
First and Second week.
Material covered: Chapter 1 sections 1.1 to 1.9.
Exercises: 1.10, 1.17, 1.21, 1.27, 1.30, 1.44, 1.52.
First HW collection: 1/28
Problems: 9, 14, 17.
Solution set for the first HW.
Chapter 2 sections 2.1, 2.2, 2.4.
Exercises: 2.10, 2.11, 2.24.
Material covered: Chapter 2 sections 2.3 and 2.5, Chapter 3 sections 3.1 and 3.2.
Exercises: 2.10, 2.11, 3.8.
Second HW collection: 2/11
Problems for Chapter 2: 4, 5, 7.
Material covered: Chapter 3 section 3.3 to 3.5 and Chapter 4 section 4.1 and 4.2.
Exercises: 3.25, 3.42, 4.18.
Problems: 4, 7.
Solution set for the second HW.
Material covered: Chapter 4 section 4.3 and 4.4 Chapter 5 section 5.1 to 5.4.
Exercises: 4.41, 5.13, 5.18, 5.30, 5.32.
The first midterm will be posted on Thursday February 25. It will cover Chapter 1 to 5.4 of the book. Here is some practice material from previous years thanks to prof. Houdré: Spring 18 and Fall 18 plus Spring 19.
Solution set for the third HW.
Solution set for the first midterm.
Material covered: Chapter 5 sections 5.5 and 5.6, Review for midterm.
Problems: 7, 11
Material covered: Chapter 6 sections 6.1 to 6.4.
Exercises: 6.14, 6.26, 6.36, 6.45, 6.55.
Material covered: Chapter 6 sections 6.5 to 6.8 and section 7.1.
Exercises: 6.61, 6.70, 6.80.
Problems: 6, 20
Material covered: Chapter 7 section 2 to 4.
Exercises: 7.10, 7.36.
>Solution set for the forth and fifth HW.
Material covered: Chapter 7 sections 5 and 6.
Exercises: 7.60, 7.75, 7.96.
Problems for chapter 7: 8, 11.
The second midterm will be on Thursday April 1st. It will cover up to Chapter 6 and concentrate on Chapter 5 and 6 of the book. Here is some practice material from previous years thanks to prof. Houdré: Spring 18 and Fall 18 and my Spring 19.
Solution set for the second midterm.
Material covered: Chapter 8 sections 1 and 2.
Exercises: 8.10, 8.11, 8.21
Material covered: Chapter 8 sections 3 to 5.
Problems for chapter 8: 5, 14