Course Description:

Mathematically rigorous, measure-theoretic introduction to probability spaces and measures, the notion of independence, and properties of random variables. The ultimate goal is to introduce, prove, and understand various forms of LIMIT THEOREMS, such as WEAK and STRONG LAW OF LARGE NUMBERS, and CENTRAL LIMIT THEOREMS.

Instructor:

Aaron Nung Kwan Yip

Department of Mathematics

Purdue University

Contact Information:

Office: MATH 432

Email: click here

Lecture Time and Place:

T, Th 1:30 - 2:45, REC 113

Office Hours:

W, F 10-11am, or by appointment

Textbook:

(All of the following are on reserve in math library.)

Probability and Measure, by Patrick Billingsley, third edition

Reference:
Probability: theory and examples, by Richard Durrett.
Probability, by Leo Breiman (also available online from library page using Purdue web-address)
Theory of Probability and Random Processes, by L. B. Koralov, Y. G. Sinai (also available online from library page using Purdue web-address)
An introduction to probability theory and its applications, v.1 and 2, by William Feller

Prerequisites:

Good working knowledge of basic mathematical analysis, at the level of MA 440-442, or MA 504. Previous experience of probability at the level of MA/STAT 519 is desirable but not absolutely necessary.

Homework:

Homeworks will be assigned (probably) bi-weekly. They will be gradually assigned as the course progresses. Please refer to the course announcement below.

• Steps must be shown to explain your answers. No credit will be given for just writing down the answers, even if it is correct.
  
  
• Please staple all loose sheets of your homework to prevent 5% penalty.
  
  
• Please resolve any error in the grading (hws and tests) WINTHIN ONE WEEK after the return of each homework and exam.
  
  
• No late homeworks are accepted (in principle).
  
  
• You are encouraged to discuss the homework problems with your classmates but all your handed-in homeworks must be your own work.

Examinations:

Test: One evening exam (date to be determined)

Final Exam: During Final Exam Week

Grading Policy:

Homeworks (50%)

Test (20%)

Final Exam (30%)

You are expected to observe academic honesty to the highest standard. Any form of cheating will automatically lead to an F grade, plus any other disciplinary action, deemed appropriate.

Course Outline:

Probability Space and Measure (Chapters 1, 2)

Integration Theory (Chapter 3)

Random Variables (Chapter 4)

Law of Large Numbers (Chapter 4)

Central Limit Theorem (Chapter 5)

Course Progress and Announcement:

(You should consult this section regularly, for homework assignments, additional materials and announcements.)

Review of inf, sup, liminf and limsup

Homework 1 Due Tuesday, Feb 5, in class (solution)

Homework 2 Due Tuesday, Feb 19, in class (solution)

Homework 3 Due Thursday, Mar 7, in class (solution)

Homework 4 For practice only, no need to hand in (solution)

Midterm Examination: March 26, Tuesday, 8-10pm, BRNG B268
(Unlimited) open notes (as long as the notes are prepared by yourself, but no copies of any textbooks).
Notes and homework solutions posted in the course webpage are fine.
No electronic devices.
(Class on Apr 9, Tuesday will be cancelled.)

An example of a past midterm

Midterm Solution

Homework 5 Due Tuesday, Apr. 16, in class (solution)

Homework 6, for practice only, no need to hand in:
[Bill]: 26.9, 15, 20, 21; 27.1, 4, 8, 11, 16, 17, 18 (solution)

Final Examination: May 1, Wednesday, 8-10am, REC 113
(Unlimited) open notes (as long as the notes are prepared by yourself, but no copies of any textbooks).
Notes and homework solutions posted in the course webpage are fine.
No electronic devices.

An example of a past final

Final Exam Solution