Spring 2021: Math 431

Spring 2021: Math 431 - Introduction to the Theory of Probability (Section 1 and Section 2)

Section 1: Online MWF 9:55 AM-10:45 AM CST (Synchronous) via Zoom (Meeting ID: 930 1223 2226, password required)

Section 2: Online (Asynchronous)

Math 431 Section 1 and Math 431 Section 2 will be taught together, there will be live lectures via Zoom at the times of the synchronous lecture, Section 1 MWF 9:55 AM-10:45 AM, but these will be recorded and uploaded. We will have one joint canvas course and Piazza.

Course Description: Math 431 is an introduction to the theory of probability, the part of mathematics that studies random phenomena. Topics covered include axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, how and when to estimate probabilities using the normal or Poisson approximation, 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.

Prerequisites: Math 234 (Calculus 3 - Functions of Several Variables).

Office hours: MWF 10:45 – 11:45 AM or by appointment

Textbook: Introduction to Probability by Anderson, Seppalainen, and Valko

Some highlights (More details are provided in the syllabus):

1. This course can be challenging if they do not keep up with the coursework. Students are expected to explain their solutions on the homework assignments and exams. A numerical answer on its own is not sufficient.

2. Attendance in lectures (or optional problem-solving sessions if any) is not mandatory, but it is highly encouraged.

3. We will be using Piazza for class discussion. Participation in Piazza is required. You should make at least one contribution every two weeks. Contributions are posts, responses, edits, followups, and comments to followups (i.e., everything).

4. Extra credit can be given to students who actively participate in their classes.

5. For both Sections 1 and 2, midterm exams and the final exam will be in the evenings:

Midterm 1: Wednesday, March 3, 2021, 5:30 pm - 7 pm

Midterm 2: Wednesday, April 7, 2021, 5:30 pm - 7 pm

Final Exam: Friday, May 7, 2021, 5:05 pm - 7:05pm

Make sure to reserve them in your calendar (Very important!). If you are unable to attend any of the exams, then you should let me know by the end of the first week. Exam attendance is mandatory.

Weekly Schedule

Week 1. Axioms of probability, sampling, review of counting, infinitely many outcomes, review of the geometric series (Sections 1.1-1.3).

Week 2. Rules of probability, random variables, conditional probability (Sections 1.4-1.5, 2.1).

Week 3. Bayes formula, independence, independent trials (Sections 2.2-2.4).

Week 4. Independent trials, birthday problem, conditional independence, probability distribution of a random variable (Sections 2.4-2.5, 3.1).

Week 5. Cumulative distribution function, expectation and variance (Sections 3.2-3.4).

Week 6. Gaussian distribution, normal approximation and law of large numbers for the binomial distribution, (Sections 3.5 and 4.1-4.2).

Week 7. Applications of normal approximation, Poisson approximation, exponential (Sections 4.3-4.5).

Week 8. Moment generating function, distribution of a function of a random variable (Sections 5.1-5.2).

Week 9. Joint distributions (Sections 6.1-6.2).

Week 10. Joint distributions and independence, sums of independent random variables, (Sections 6.3-7.1).

Week 11. Expectations of sums and products, exchangeability (Sections 7.2-8.1).

Week 12. variance of sums, Sums and moment generating functions, (Sections 8.2-8.3).

Week 13. covariance and correlation, Markov and Chebyshev inequalities, (Sections 8.4-9.1).

Week 14. Law of large numbers, central limit theorem, Review (Sections 9.2-9.3).

References.

1. Video: Prediction By The Numbers

https://youtu.be/9OIel5NUG7Q?t=706

2. Syllabus by Benedek

http://www.math.wisc.edu/~valko/courses/431/431.html