Math 5010/6805: Introduction to Probability
Lecture 2
Instructor: Chris Janjigian
Lecture: Monday and Wednesday 6:00-7:30 PM LCB 215
Office Hours: M 1:00-3:00 PM JWB 130 (subject to change) or by appointment
Piazza signup link.

Pre-requisites and course expectations


This course requires a solid foundation in Calculus I, II, and III and does not include a refresher on calculus. If you have not had a chance to think about calculus in some time, I would recommend finding a calculus textbook and practicing computation of both derivatives and integrals during the first few weeks of the semester. You are expected to be able to compute derivatives and (some) integrals of polynomials, logarithms, exponentials, and trigonometric functions. This course will involve computing multidimensional integrals and will involve multidimensional change of variables.

Attendance is not mandatory, but be aware that it will be difficult to keep up with the course without attending class.

We will cover some the classical aspects of probability theory including combinatorial problems, discrete and continuous random variables, distributions, independence, conditional probability, expected value and moments, law of large numbers, and the central limit theorem. There is a tentative schedule below, which will be adjusted as the semester progresses.

This course is three credit hours and serves as a QRQI course (quantitative reasoning-Math, quantitative reasoning-statistics/logic, and quantitative intensive). Expect a mixture of "problem solving" and "theory and concepts" problems. Based on previous semesters, the ratio will likely be about two thirds will be of the former type and one third will be of the latter. "Theory and concepts" problems will require conceptual understanding and not just memorization.

Please post questions about homework or the quizzes to Piazza, so the rest of the class can see the answer. Feel free to post anonymously if you would prefer to not be identified.

If you are struggling in the class or would like to discuss the course material, please come to my office hours or schedule a time to meet with me. You are also invited to visit the tutoring center, which is located in the T. B. Rushing Undergraduate Student Center in the basement of LCB.

Textbook

We will largely be following the department course notes by Davar Khoshnevisan and Firas Rassoul-Agha, which are available for free online here. We will cover two or three of the lectures in these notes in most weeks. Due to time constraints, some material may be covered in less detail than in the notes. Homework will either be issued out of these notes or will be available as a PDF on this website. This is the only text that is required for the course, but I would recommend using supplementary text(s) as well.

I recommend picking up a copy of Introduction to Probability by Anderson, Seppalainen, and Valko, ISBN: 9781108415859. The book is well-written with lots of solved exercises and is presented in a similar way to how I will present the class. I will include suggested (but not required) readings from this text as the course progresses.

These notes are based in part on the textbook Basic Probability Theory by Robert Ash (available for free on the author's website here). This text is not required for the course, but may be helpful.

Grading


See below for the breakdown of course grades. Grading will be based on weekly homework and quizzes along with two midterm examinations and one final examination. Homework will be graded for effort. It will primarily be to confirm that you are keeping up with the coursework.

Quizzes will occur on Mondays and will be based on the material covered in the previous week. These will be your primary method of obtaining feedback in the course other than the exams. Except in exceptional circumstances, there will be no make-up quizzes. To accommodate unavoidable absences, your lowest three quiz grades will be dropped.

There will be two midterm examinations and one final. On the midterms and the final students will be evaluated both on the correctness of their arguments and the final answer, with an emphasis on the correctness of the argument. The tentative dates for the exams are below. It is important to show your work as efficiently as possible. The examinations are designed to be the real evaluation of your work in the course. If your grade on the three exams without including quizzes and homework is higher than your grade including quizzes and homework, then that will be your grade in the course.


Category
Percent
Homework 5
Quiz 15
Midterm 25
Midterm 25
Final 30
Total 100


Worksheets and Homework

Week
Chapters
Topics
Homework/Quiz/Exam
Week 1:
1/8, 1/10
Lecture Notes: 1,2,3,4 (abridged), 5
ASV: 1.1, 1.4
Introduction, sample spaces, and rules of probability, combinatorics Homework: Lecture notes 1.1, 1.3, 1.4, 1.6, 2.1, 3.1, 4.1, 4.4
Solutions: Homework 1
Week 2:
1/17
Lecture Notes: 5,6,7
ASV: 1.2
Combinatorics: (un)ordered sampling with(out) replacement, permutations, combinations Homework:5.1, 6.2, 6.6, 6.8, 6.9, 7.1, 7.2, 7.3 (only do the combinatorial proof)
Quiz: Quiz 1
Solutions: Homework 2
Quiz 1 Solutions
Week 3:
1/22, 1/24
Lecture Notes: 8,9
ASV: 2.1-2.4
Conditional probability, Bayes' rule, independence Homework: 8.2, 8.3, 8.7* (see CANVAS for an edit), 8.8, 9.1, 9.3, 9.5.
Quiz:Quiz 2
Solutions: Homework 3
Quiz 2 Solutions
Week 4:
1/29, 1/31
Lecture Notes: 10,11,12
ASV: 2.5,4.4
Random variables examples: Bernoulli, Binomial, Poisson, negative binomial Homework:10.1, 11.2, 11.3, 11.4, 11.5, 11.6, 12.1, 12.2
Quiz:Quiz 3
Solutions: Homework 4
Quiz 3
Week 5:
2/5, 2/7
Lecture Notes: 13,14,15,16
ASV:3.1,3.2
Cumulative distribution functions; continuous random variables: uniform, exponential, normal, cauchy Homework: 13.1, 14.1, 14.2, 14.4, 14.5
Quiz:Quiz 4
Solutions: Homework 5
Quiz 4
Week 6:
2/12, 2/14
Lecture Notes: 16
ASV:
Continuous distributions; exam Exam 2/14
Week 7:
2/21
Lecture Notes: 17,18,19,20
ASV: 4.1,4.2, 4.3, 5.2,5.3
Bernoulli and De-Moivre-Laplace theorems, Continuous random variables continued, functions of random variables Homework: 19.1, 19.2, 19.3, 19.5, 19.6
Solutions: Homework 6
Week 8: 2/26, 2/28 Lecture Notes: 21 22, 23, 24
ASV: 5.2, 5.3, 6.1, 6.2
Functions of random variables, generating random variables, continuous joint distributions, marginal distributions Homework: 21.1, 21.2, 21.3, 24.1, 24.2, 24.3, 24.4, 24.5
Quiz:Quiz 5
Solutions: Homework 7
Quiz 5 Solutions

Week 9: 3/5, 3/7 Lecture Notes: 25, 26, 27
ASV: 6.2, 6.3, 6.4, 3.3
Functions of a random vector, expectation for discrete random variables Homework: 25.1, 25.2, 25.3, 25.7, 25.10, 27.1, 27.2
Quiz:Quiz 6
Solutions: Homework 8
Quiz 6 Solutions
Week 10: 3/12, 3/14 Lecture Notes: 28, 29, 30
ASV:
Expectation and continuous variables, properties of expectations, variance Homework: 29.1, 29.2, 29.3, 30.1, 30.2, 30.3
Quiz:
Solutions: Homework 9
Week 11: 3/26, 3/28 Lecture Notes: 31
ASV:
Variance continued; exam Exam 3/28
Week 12: 4/2, 4/4 Lecture Notes: 32,33
ASV: 8.4, 9.1, 9.2
Covariance, indicator functions, the law of large numbers Homework: Homework 10
Solutions: Homework 10 Solutions
Week 13: 4/9, 4/11 Lecture Notes:
ASV: Chapter 10
Conditioning Homework: Homework 11
Quiz:
Solutions: Homework 11 Solutions
Week 14: 4/16, 4/18 /td> Lecture Notes: br/> ASV: Chapter 5 Moment generating functions and moments, the central limit theorem Homework: Homework 12
Quiz:
Solutions: Homework 12 Solutions
Week 15: 4/23 Lecture Notes: 38
ASV:
The central limit theorem, review Quiz:
Solutions:
Examinations

Exam
Chapters covered
Questions
Solutions
Midterm 1: Lecture Notes: 1-17

Midterm 2: Lecture Notes: 18 - 31

Final: Comprehensive