6.436J/15.085J: Course Information

(Fall 2015)

(version of 8/28/15)

Description

This is a course on the fundamentals of probability geared towards first or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in 6.041/6.431 (sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, limit theorems) but at a faster pace and in a lot more depth. There is also a number of additional topics such as: language, terminology, and key results from measure theory; interchange of limits and expectations; multivariate Gaussian distributions; deeper understanding of conditional distributions and expectations.

Intended audience

The course is geared towards students who need to use probability in their research at a fairly sophisticated level, e.g., to be able to read the research literature in communications, stochastic control, machine learning, queueing, etc., and to carry out research involving precise mathematical statements and proofs. One of the objectives of the course is the development of mathematical maturity.

Prerequisites

While the only formal prerequisite is 18.02, the course will assume some familiarity with elementary undergraduate probability and some mathematical maturity. A course in analysis will be helpful. It is not required, but be prepared to work harder if you have not had it. 

Homework, exams, etc.

• There will be about 11, more or less equally spaced, homework sets. Together with the TA's feedback, they will count for 20% of the final grade. Homework solutions will be handed out on the day that the homework is due. Late homework will be heavily discounted. 
• The 11th homework will not be graded. Out of the remaining 10, we will only count the best 9.
• There will be a 2-hour quiz, tentatively scheduled for the evening of Tuesday, October 19, 7:30-9:30pm, which will count for 25% of the final grade.
• There will be another 90-minute quiz, tentatively scheduled for the evening of Tuesday, November 10, 7:30-9:00pm, which will count for 20% of the final grade.
• No homework will be due during a quiz week.
• There will be a comprehensive final exam, during final's week, which will count for 35% of the final grade.

Recitation

• There will be two recitation sections, on Fridays at 2 and 3. In order to balance TA load, you will submit your preferences at the first lecture, and will assign students to sections accordingly. If you miss the first lecture, please email your availability to the TAs.
• Recitation will typically review some of the lecture material, clarify some concepts, present some examples, and solve problems.
  

Policy on collaboration and academic honesty

• You may interact with fellow students when preparing your homework solutions. However, at the end, you must write up solutions on your own. Duplicating a solution that someone else has written (verbatim or edited), or providing solutions for a fellow-student to copy is not acceptable. If you do collaborate on homework, you must cite, in your written solution, your collaborators. Also, if you use sources other than assigned readings in one of your solutions, e.g., a friendly "expert", or another text, be sure to cite the source. There is no penalty for such collaboration or use of other sources, as long as it is disclosed. 
• In general, we expect students to adhere to basic, common sense concepts of academic honesty. Presenting somebody else's work as if it were your own, or cheating in exams, is unacceptable, and appropriate disciplinary procedures will be followed.

Readings 

You will only be responsible for the material contained in lecture notes and other handouts. All of these materials will be available on the class Stellar site.

However, the lecture notes are somewhat sparse, with few examples. For additional reading and examples, you can use the books listed below. The first four are on reserve at the Barker Engineering Library. Furthermore, the book by Florescu is available online through the MIT library site. 

Recommended:

This book covers the core materials  at more or less the same level as this class:

I. Florescu, Handbook of probability, Wiley, 2015.

This book is also pretty close to this class in terms of level and coverage, although some students have found it to not be a great match in terms of style and emphasis. Nevertheless, it can be a nice complement:

G. R. Grimmett and D. R. Stirzaker, Probability and Random processes, Oxford University Press, 3rd edition, 2001, ISBN 0198572220.

The course syllabus is somewhat parallel to the 6.041/6.431 syllabus, although at a different level. The 6.041/6.431 textbook can be useful for reviewing a more elementary version of the material:

D. P. Bertsekas and J. N. Tsitsiklis, Introduction to Probability,  2nd Ed., Athena Scientific, 2008, ISBN 978-1-886529-23-6.

For those who have some analysis background and wish to see the “full story,” including complete proofs and much more:

A concise, crisp, yet complete and rigorous treatment of the theoretical topics in this class,  although at a higher mathematical level:

D. Williams, Probability with Martingales, Cambridge University Press, 1991, ISBN 0521406056.

A very well written, and accessible development of basic measure-theoretic probability:

M. Adams and V. Guillemin, Measure Theory and Probability, Birkhauser, 1996.

An encyclopedic, very comprehensive reference:

P. Billingsley, Probability and Measure, 3rd Edition, Wiley, 1995.

Very comprehensive, but quite demanding compared to the level of this class:

R. Durett, Probability: Theory and Examples, 4th Edition, Cambridge University Press, 2010, ISBN:0534424414.