**Course Announcement**

**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

**Course Syllabus**

**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

**Homework**- Homework #1, due 09/01/2010

- Homework #2, due 09/10/2010

- Homework #3, due 09/17/2010

- Homework #4, due 09/24/2010

- Homework #5, due 10/01/2010

- Homework #6, due 10/08/2010

- Homework #7, due 10/15/2010

- Homework #8, due 10/22/2010

- Homework #9, due 10/29/2010

- Homework #10, due 11/05/2010

- Homework #11, due 11/12/2010

- Homework #12, due 11/19/2010

- Homework #13, due 11/24/2010

- Homework #14, due 12/01/2010

- Homework #1, due 09/01/2010
**Relevant Course Materials**- Contents of the textbook

- Possible typos of the textbook (Note: there are several different prints of the textbook.)

- Gerald B. Folland,
*Real Analysis: Modern Techniques and Their Applications*, John Wiley & Sons, 1999 -- textbook of MATH 534, use it as a reference book

- Patrick Billingsley,
*Probability and Measure*, 3rd edition, John Wiley & Sons, 1995 -- a good reference book

- Jeffrey S. Rosenthal,
*A First Look at Rigorous Probability Theory*, 2nd edition, World Scientific Publishing Company, 2006 -- a good reference book for readers without sufficient mathematical background (thank Richard Schwinn for recommendation)

- Bruce K. Driver,
*Math 280 Probability Lecture Notes*, 2009-2010 -- good for readers who want more relevant readings (thank Yan Xing for recommendation)

- Contents of the textbook

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