# Math 461: Probability Theory (Spring 2023)

An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, random variables and probability distributions, expectation, binomial distribution, Poisson and normal approximation, generating functions and the Central Limit Theorem. We will cover most of the material in the first eight chapters of the textbook.

• Chapter 1, Combinatorial Analysis, 4 hours
• Chapter 2, Axioms of Probability (omit sections 2.6, 2.7), 5 hours
• Chapter 3, Conditional Probability and Independence (omit section 3.5), 5 hours
• Chapter 4, Random Variables (omit subsection 4.8.4), 6 hours
• Chapter 5, Continuous Random Variables (omit subsections 5.5.1, 5.6.2, 5.6.4), 5 hours
• Chapter 6, Jointly Distributed Random Variables (omit sections 6.6, 6.8), 5 hours
• Chapter 7, Properties of Expectations (omit subsections 7.2.1, 7.2.2, 7.7.1, and section 7.9), 7 hours
• Chapter 8, Limit Theorems (omit sections 8.4, 8.5, 8.6), 3 hours

Instructor Partha Dey
Office 35 CAB
ContactBy email with subject line: "Math 461:"
Class TR 9:30-10:50am at 245 Altgeld Hall.
TextbookSheldon Ross, A First Course in Probability, 9th Edition, ISBN: 9780321794772.

It is okay to use a different edition for studying.

The books Introduction to Probability Theory by Hoel, Port and Stone; and Probability and Statistics by Morris DeGroot and Mark Schervish are optional as reference texts.

The book Introduction to Probability by C. M. Grinstead and J. L. Snell is available free online.

You can also freely view the ebook Introduction to Probability by Blitzstein & Hwang.
Prerequisite Math 241 or the equivalent. We will use important topics from calculus, such as infinite series with positive terms (most calculations involve the geometric series and series derived from it), improper integrals and double integrals (change of variables formula, manipulating Gaussian integrals).
DRESTo obtain disability-related academic adjustments and/or auxiliary aids, students should contact both the instructor and the Disability Resources and Educational Services (DRES) as soon as possible. You can contact DRES at 1207 S. Oak Street, Champaign, (217) 333-1970, or via e-mail at disability@illinois.edu.
Grading Policy Homework: 20% of the course grade. Homework will be assigned weekly on Thursdays on Canvas, to be submitted at the start of next Thursday lecture or earlier in Canvas.

Solving a lot of problems is an extremely important part of learning probability. You are encouraged to work together on the homework, but I ask that you write up your own solutions and turn them in separately. Late homework will not be graded. If for some reason you've done a homework but can't turn it in online, send it via email before class. Because of this strict policy on late homework, I will drop your lowest score. Please talk to the instructor in cases of emergency.

Quiz: 15% will depend on weekly quizzes on Prairielearn. Problems will be based on that week’s homework material. I will drop your lowest quiz score.

Midterm: 30%=2 x 15% will depend on two in-class midterm exams on (tentatively) Tuesday, Feb 28, 2023 and Tuesday, April 4, 2023. Each exam will be technically comprehensive, but emphasizing recent material up to the most recent graded and returned homework assignment. Exam problems will be similar to homework problems.

Final: 35% will depend on a final exam on (tentatively) TBA. It will cover the most important topics of the whole course, emphasizing recent material somewhat.
Exam PolicyMake-up exams will be given only for medical or other serious reasons. If you discover that you cannot be at an exam, please let me know as soon as possible, so that we can make other arrangements. You must work completely on your own during exams (and any quizzes). I make my exams fair and similar to homework, so as long as you use the resources provided, you should do fine. If you have difficulties of any kind or fall behind in the course, please come talk to me as soon as possible.
Grading scaleFinal scores will be converted to letter grades beginning with the following scale:
 A A- B+ B B- C+ C D 93 90 87 83 80 77 70 60
As for a curve, these cutoffs might be adjusted, but only in the downward direction (to make letter grades higher).

 Week Date Due Content 1 Tu Jan 17 Principle of counting, Permutation, Sec 1.1-1.3 Th Jan 19 Combinations, Binomial and Multinomial Thms. Sec 1.4-1.6 2 Tu Jan 24 Axioms of probability, Sec 2.2, 2.3 Th Jan 26 HW 1 Propositions, Equally likely outcomes, Sec 2.4, 2.5 3 Tu Jan 31 Conditional probability, Sec 3.2 Th Feb 2 HW 2 Bayes Rule, Sec 3.3 4 Tu Feb 7 Independent events and trails, Sec 3.4 Th Feb 9 HW 3 Gambler's ruin, Simpson's Paradox, RVs, Sec 4.1 5 Tu Feb 14 Discrete rvs., Expectation, Sec 4.2-4.4 Th Feb 16 HW 4 Variance. Binomial rv, Sec 4.5-4.6 6 Tu Feb 21 Poisson r.v., Expected value of sums of rvs, Sec 4.7, 4.9 Th Feb 23 HW 5 First Midterm Review 7 Tu Feb 28 MT 1 First Midterm Exam. Th Mar 2 Geometric, Negative Binomial, CDFs, Sec 4.8, 4.10 8 Tu Mar 7 Continuous rvs and Expectations, Sec 5.1-5.2, 5.7. Th Mar 9 HW 6 Uniform and Normal rvs, Sec 5.3-5.4. 9 Tu Mar 21 Normal, Exponential and Gamma rvs, Sec 5.4-5.6. Th Mar 23 HW 7 Joint distributions, Sec 6.1. 10 Tu Mar 28 Independent rvs, Sec 6.2. Th Mar 30 HW 8 Second Midterm Review. 11 Tu Apr 4 MT 2 Second Midterm Exam. Th Apr 6 Sums, Conditional distributions, Sec 6.3-6.5. 12 Tu Apr 11 Functions of rvs, Sec 6.7. Th Apr 14 HW 9 Expectation of rvs, Correlation, variance, Sec 7.2, 7.4. 13 Tu Apr 18 Conditional expectation, mgf, Sec 7.5. Th Apr 20 HW 10 MGF, Markov and Chebyshev Ineq, Sec 7.7, 8.2. 14 Tu Apr 25 Central Limit Theorem, Sec 8.3. Th Apr 27 HW 11 SLLN, Jensen's ineq, Sec 8.4-8.5. 15 Tu May 2 HW 12 Final Exam review. Last day of class. 16 May Final Exam.