MAT 1128 Topics in Probability (Graduate course): Fall 2013

Important announcement: the final exam will take place on Monday December 16 from 1-4pm in the mathematics exam room BA 6183 (6th floor of Bahen).

Course description

You can download the course syllabus here.

MAT 1128 is a course designed for Master’s and Ph.D. level students in statistics, mathematics, and other departments, who are interested in a rigorous, mathematical treatment of probability theory using measure theory. Specific topics to be covered include: probability measures, the extension theorem, random variables, distributions, expectations, laws of large numbers, Markov chains. If time permits we may cover some basic aspects of ergodic theory.

The course takes the place of STA 2111 for Mathematics graduate students and will prepare students for Probability II (STA 2211) and the Qualifying Exam in May.

Students should have a strong undergraduate background in Real Analysis, including calculus, sequences and series, elementary set theory, and epsilon-delta proofs. Some previous exposure to undergraduate-level probability theory is also recommended.

Assessment

There will be a mid-terms as well as a final exam. In addition there will be roughly biweekly assignments that will be marked and returned. The final exam will count for half of your mark, the rest will be made up by midterm (30%) and assignments (20%).

The mid-term will take place during class on Wednesday October 23.

Handouts

From time to time I will post handouts on supplementary material.

• Handout on measure theory: some facts and concepts that you will need to know for this course. These notes may be slightly extended as we move through the course.
  
• List of topics for the midterm test.
  
• List of topics for the final exam.
  
• Some extra problems for the final exam.
  

Exercises

I will regularly post exercises here. Most will be for you to test your understanding of the course. You are encouraged to work through these and come and ask me if you have problems. Some will be for handing in to me, and you will receive these returned with a grade.

• Exercise sheet 1: exercises on measure theory. This is for your own benefit only and does not contain any assessed questions.
  
• Exercise sheet 2: exercises on integration and convergence of random variables. This exercise sheet does not contain any assessed questions.
  
• Exercise sheet 3: exercises on expectation and convergence theorems. This exercise sheet does not contain any assessed questions, but you are encouraged to hand in your solutions for grading by Monday, September 30.
  
• Exercise sheet 4: (corrected version) exercises on independence. This is the first assignment that counts towards your course credit. Please submit answers before the lecture on Wednesday, October 9. If you are auditing this course you are still encouraged to submit your answers in order to get feedback.
• Exercise sheet 5: exercises on weak laws of large numbers and liminf and limsup for sets. This is the second assignment that counts towards your course credit. Please submit answers before the lecture on Wednesday, October 30. If you are auditing this course you are still encouraged to submit your answers in order to get feedback.
• Exercise sheet 6: (typo corrected) exercises on weak convergence and characteristic functions. This is the third assignment that counts towards your course credit. Please submit answers before the lecture on Wednesday, November 13. If you are auditing this course you are still encouraged to submit your answers in order to get feedback.
• Exercise sheet 7: exercises on the CLT and random walks. This is the fourth and last assignment that counts towards your course credit. Please submit answers before the lecture on Wednesday, November 27. If you are auditing this course you are still encouraged to submit your answers in order to get feedback.

Course text

The text book for the course will be Probability: Theory and Examples by Rick Durrett, fourth edition, published by Cambridge University Press. You can download a pdf copy of an earlier edition on his web site