Probability Theory (Math W4155) - Spring 2015

The purpose of this course is to provide a mathematically rigorous introduction to the basic notions and results of probability theory. We shall discuss probability spaces and random variables, measure theory and integration, probabilistic concepts such as independence, conditional expectation or moments, fundamental results like the law of large numbers and the central limit theorem, and basic models such as random walks and Markov chains.

Instructor: Marcel Nutz
Office hours: Tuesdays: by appointment. Thursdays: 9:55-10:45am, Room 618 Math
Meeting time: Tuesdays and Thursdays, 8:40-9:55am
Room: 520 Math
Teaching Assistant: Cameron Bruggeman
Recitation: TBA
Help with the course material is also available at the Mathematics Help Room

General Information

Prerequisites: Familiarity with basic notions of analysis and rigorous proofs. Official prerequisites are: Introduction to Modern Analysis (Math W4061) or Complex Variables (Math V3007).
Texts: No text is required. Some recommended texts:

J. Jacod and P. Protter, Probability Essentials, 2nd edition, 2004.
S. Resnick, A Probability Path, 1999.

D. Stirzaker, Elementary Probability, 2nd edition, 2003.

Grading: The homework and midterm will each count for 20% of the final grade, and the final exam will count for the remaining 60%.
Midterm Examination: There will be one midterm, on March 3, during class time. Students will be excused from the midterm only because of serious illness or another emergency of similar gravity; a note from a doctor or from a Dean will be required.
Final Examination: The final exam is projected for May 14, 9:00am-noon. The University will announce an official Final Examination schedule later in the semester. All students must take the final at the time scheduled by the University. No exception will be made.
Conflicts: If you have a conflict with any of the exams (for example, due to a religious holiday), please contact the instructor as soon as possible.
Written work must be neat and legible to receive consideration. You must explain your work in order to obtain full credit.
Disabilities: Students who may need disability-related accommodations should contact the instructor as soon as possible. Also, stop by the Office of Disability Services to register for support services.
Calculators are not required for this course, and no calculators or other electronic devices will be allowed during any of the exams.

Exercises

There will be weekly homework assignments, generally assigned on Tuesdays and due the following Tuesday by 10:00am in the box next to 417 Math. Late homework will not be accepted. Graded homework will delivered to the tray next to 520 Math, generally by Thursday. Your lowest two homework scores will be dropped, and it is expected that this will take care of any homework you may fail to hand in due to illness, travel, bad luck, and so on. You are encouraged to discuss the homework with fellow students, but you must write your solutions individually. Exercises and Solutions are posted on Courseworks.

Syllabus

The syllabus is tentative and may change during the semester.
Please consult the Academic Calendar for important dates such as the last day to drop the class.

I. Basics

1. Probability Spaces S 1.1-14; R 1.1-9, 2.1
2. Discrete Random Variables S 4.1-3
3. General Random Variables

a) Transformation of Probability Spaces R 3.1-3
b) Uniqueness of Probability Measures R 2.2-3
c) Probability on the Real Line S 4.1-2,7.1
d) Expectation S 4.3, 8.5; R 5.1-6

4. Some Probabilistic Inequalities S 4.6; R 6.6

II. Dependence and Independence

1. Conditional Probability and Distribution S 2.1-13
2. Independence S 5.2, 8.3; R 4.1-2, 4.5
3. (In)dependence and Correlation S 5.3, 5.11
4. Product Measure and Fubini's Theorem S 5.1-4, 8.1-4, 8.17; R 5.7-9
5. Random Walk and Gambling S 2.11, 5.6, 5.18, 5.20

III. Limit Theorems

1. The Weak Law of Large Numbers S 5.8; R 6.1-3, 7.2
2. The Strong Law of Large Numbers R 7.5
3. The Central Limit Theorem S 7.5, 8.9

IV. Markov Chains

1. The Markov Property S 9.1-2
2. Hitting Times S 9.3
3. The Strong Markov Property
4. Recurrence and Transience S. 9.5
5. Convergence to Equilibrium S. 9.4-5
6. Applications