Fundamentals of probability

Lectures: 2 sessions / week, 1.5 hours / session

Recitations: 1 session / week, 1 hour / session

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 6.431 (sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, limit theorems) but at a faster pace and in more depth. There are 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.

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 would be helpful but is not required.

The course is geared towards students who need to use probability in their research at a reasonably sophisticated level (e.g., to be able to read the research literature in communications, stochastic control, machine learning, queuing, 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.

You will only be responsible for the material contained in lecture notes and other handouts. However, the lecture notes are somewhat sparse, with few examples. For additional reading and examples, you can use the following books.

This book comes closest to this course in terms of level and coverage:

Grimmett, Geoffrey, and David Stirzaker. Probability and Random Processes. 3rd ed. Oxford University Press, 2001. ISBN: 9780198572220.

For a concise, crisp, and rigorous treatment of the theoretical topics to be covered (although at a somewhat higher mathematical level):

Williams, David. Probability with Martingales. Cambridge University Press, 1991. ISBN: 9780521406055.

The course syllabus is a proper superset of the 6.041/6.431 syllabus. For a more accessible coverage of that material:

Bertsekas, Dimitri, and John Tsitsiklis. Introduction to Probability. 2nd ed. Athena Scientific, 2008. ISBN: 9781886529236.

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

Adams, Malcolm, and Victor Guillemin. Measure Theory and Probability. Birkhäuser Boston, 1996. ISBN: 9780817638849

A rather encyclopedic and most comprehensive reference:

Billingsley, Patrick. Probability and Measure. 3rd ed. Wiley, 1995. ISBN: 9780471007104.

Comprehensive, but significantly more advanced:

Durrett, Rick. Probability: Theory and Examples. 4th ed. Cambridge University Press, 2010. ISBN: 9780521765398.

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