Time: Monday, Wednesday, Friday at 2:00 p.m. - 2:50 p.m.
Location: Taft Hall 313
Instructor: Jie Yang
Office: SEO 513
Phone: (312) 413-3748
E-Mail: jyang06 AT math DOT uic DOT edu
Office Hours: Monday, Wednesday, Friday at 12:50 p.m. - 1:50 p.m. (or by appointment)
Textbook: Sidney I. Resnick,
A Probability Path, Birkhäuser, 1999.
Content: Sets and events; probability spaces; random variables, elements and measurable maps; independence; integration and expectation; convergence concepts
Prerequisite: MATH 534 or consent of instructor
Homework:
Turn in every Wednesday before class;
half of the grade counts for completeness;
half of the grade counts for correctness of one selected problem.
Short Presentations:
Each student is required to do two short presentations during the course period.
One presentation should be done in front of the whole class.
The other one may be done during the office hour.
Each presentation may last up to 15 minutes.
The topics of presentations may come from the optional part of homework assignments.
Grading: Homework 50%, presentations 25% each
Grading Scale: 90% A , 75% B , 60% C , 30% D
| WEEK | SECTIONS | BRIEF DESCRIPTION |
| 08/23 - 08/27 | 1.1; 1.2; 1.3 | Introduction; Basic Set Theory; Limits of Sets |
| 08/30 - 09/03 | 1.4; 1.5; 1.6, 1.7; | Monotone Sequences; Set Operations and Closure; Sigma-field Generated by a Given Class, Borel Sets on the Real Line |
| 09/06 - 09/10 | Holiday; 1.8; 2.1 | Comparing Borel Sets; Basic Definitions and Properties of Probability Spaces |
| 09/13 - 09/17 | 2.1; 2.2; 2.2 | Basic Definitions and Properties of Probability Spaces; More on Closure |
| 09/20 - 09/24 | 2.3; 2.4; 2.4 | Two Constructions; Constructions of Probability Spaces |
| 09/27 - 10/01 | 2.4; 2.5; 3.1 | Constructions of Probability Spaces; Measure Constructions; Inverse Maps |
| 10/04 - 10/08 | 3.2; 3.2; 3.3 | Measurable Maps, Random Elements, Induced Probability Measures; Sigma-Fields Generated by Maps |
| 10/11 - 10/15 | 4.1, 4.2; 4.3; 4.4 | Basic Definitions of Independence, Independent Random Variables; Two Examples of Independence; More on Independence: Groupings |
| 10/18 - 10/22 | 4.5; 4.5; 4.6 | Independence, Zero-One Laws, Borel-Cantelli Lemma |
| 10/25 - 10/29 | 5.1; 5.2; 5.2 | Preparation for Integration; Expectation and Integration |
| 11/01 - 11/05 | 5.3; 5.4; 5.5 | Limits and Integrals; Indefinite Integrals; Transformation Theorem and Densities |
| 11/08 - 11/12 | 5.6; 5.7; 5.8 | Riemann vs Lebesgue Integral; Product Spaces; Probability Measures on Product Spaces |
| 11/15 - 11/19 | 5.9; 6.1; 6.2 | Fubini's Theorem; Almost Sure Convergence; Convergence in Probability |
| 11/22 - 11/26 | 6.3; 6.3; Holiday | Connections between a.s. and i.p. Convergence |
| 11/29 - 12/03 | 6.4; 6.5; 6.6 | Quantile Estimation; Lp Convergence; More on Lp Convergence |