### Course Meeting Times

Lectures: 1.5 hours / session; 2 sessions / week

Recitations: 1 hour / session; 1 session / week

### Course Prerequisites

While the only formal prerequisite is *18.02 Multivariable Calculus*, 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.

### Course 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 Probabilistic Systems Analysis and Applied Probability* (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.

### Textbooks

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.

#### Highly recommended:

This book comes closest to this class in terms of level and coverage.

- G. R. Grimmett and D. R. Stirzaker.
*Probability and Random Processes.*Oxford University Press, 3rd edition, 2001. ISBN: 9780198572220.

#### Recommended:

An excellent and complete recent monograph written by a leading expert on stochastic processes, martingales and Markov processes:

- E. Çinlar.
*Probability and Stochastics.*Springer, 2011. ISBN: 9780387878584.

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

- D. Williams.
*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:

- D. P. Bertsekas and J. N. Tsitsiklis.
*Introduction to Probability.*2nd Ed., Athena Scientific, 2008. ISBN: 9781886529236.

Chapter 1 is available online.

#### Other references:

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

- M. Adams and V. Guillemin.
*Measure Theory and Probability.*Birkhauser, 1996. ISBN: 9780817638849.

A rather encyclopedic and most comprehensive reference:

- P. Billingsley.
*Probability and Measure.*3rd Edition, Wiley, 1995. ISBN: 9780471007104.

Comprehensive, but significantly more advanced:

- R. Durrett.
*Probability: Theory and Examples.*3rd Edition, Duxbury Press, 2004, ISBN: 9789534424411.

### Grading

Activities | Percentages |
---|---|

Homework | 30% |

Midterm | 30% |

Final Exam | 40% |

There will be 12 equally spaced homework sets. Together with the TA’s feedback, they will count for 30% 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 12th homework will not be graded. Out of the remaining 11, we will only count the best 10.

There will be a 2-hour midterm exam which will count for 30% of the final grade.

There will be a comprehensive final exam, during finals week, which will count for 40% of the final grade.