Probability Theory

STAT7032, Spring semester, 2017

Instructor: Yizao Wang
Email: yizao.wang@uc.edu
Office: 4302 French Hall
Office Hours: W2:40-4pm, other time by appointment
Class meeting: MWF 11:10am - 12:05pm, Room 4206 French Hall (Seminar Room)

Textbook

• Lecture notes updated at blackboard.
• (Recommended) Probability: Theory and Examples, 4th Edition (Cambridge Series in Statistical and Probabilistic Mathematics) by Rick Durrett.
  A PDF version from the website of the author.
  

Course description

We will cover the following materials: measure-theoretic foundations of probability; laws of large numbers; weak convergence; characteristic functions; central limit theorems. The course is required for the Statistics/Probability Preliminary Exam (probability part).

Pre-req: Advanced Calculus (MATH 6001/6002) or equivalent.

Grades

• Grading: homework 10%, midterm 1 20%, midterm 2 20%, final 40%, participation 10%.

Tentative Schedule and Homework

• Week 1 (Jan 9): 1.1 Measure spaces, 1.2 Measurable functions. HW1 due Jan 20 Fri
• Week 2 (Jan 16): No class on MLK. 1.3 Integration.
• Week 3 (Jan 23): 1.4 Properties of integration.
• Week 4 (Jan 30): 1.5 Product spaces, Fubini's theorem. 2.1 Random variables as measurable functions. HW2 due Feb 3 Fri
• Week 5 (Feb 6): 2.2 Distributions, 2.3 Expected values. 2.4 Important examples of distributions. HW3 due Feb 10 Fri
• Week 6 (Feb 13): 2.5 Independent random variables. HW4 due Feb 22 Wed Midterm 1, Feb 15 (Wed) in class, materials from Chapters 1 and 2.1-2.3. Midterm 1 2016
• Week 7 (Feb 20): 3.1 Convergence in probability and in $L^p$. 3.2 $L^2$-convergence for partial sums of random variables. HW5 due Feb 29 Wed
• Week 8 (Feb 27): 3.3 Triangular arrays. 3.4 Weak law of large numbers. HW6 due Mar 8 Wed
• Week 9 (Mar 6): 3.5 Borel-Caltelli Lemmas. 3.6 Strong law of large numbers. No class on Mar 10 Fri, Probability Colloquium on Mar 9 Thur instead, HW7 due Mar 22 Wed
• Week 10 (Mar 13): Spring break
• Week 11 (Mar 20): 3.7 Convergence of random series.
• Week 12 (Mar 27): 4.1 Weak convergence. HW8 due April 7 Fri. Midterm 2, Mar 31 (Fri) in class, materials from Chapter 3. Midterm 2 2016
• Week 13 (Apr 3): 4.2 Central limit theorem. 4.3 A multivariate central limit theorem. HW9 due Apr 14, Fri
• Week 14 (Apr 10): 4.4 Random walks and Brownian motion. 4.5 Poisson convergence.
• Week 15 (Apr 17): Review. No class on Apr 21, Fri. Probability Colloquium on Apr 20 Thur instead.
• Week 16 (Apr 24): Final, April 24 (Mon), 9:45-11:45am, Seminar Room. Final 2016

Sections 4.4, 4.5: expository lectures, not covered by final exam and the prelim.

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