Lecturer: Marco Lenci. Office: Math, 5D-148. Tel.: xxx-xxxx. E-mail: xxxx@math.sunysb.edu. Office hours: Th 2:30-3:30 and by appointment.
This will be a course on the mathematical formulation of probability theory. While the purpose of the class is to give the students a set of ideas and techniques that find concrete application in the real world, the methods will be those of rigorous mathematics, mainly measure theory.
This approach is necessary if we want to touch upon topics, such as large deviations, random processes, stochastic differential equations, that are generally regarded to go beyond the elementary level. The competence on such subjects seems to be in increasing demand, in the sciences as well as in the job market.
The class is aimed at math, applied math, physics, computer science majors that are comfortable with the notion of a mathematical proof. Besides that, the official prerequisite is a course in Analysis, such as MAT 320 or equivalent (but we can possibly be lenient on this).
As regards the syllabus, a great deal of input is expected from the students, who are welcome to express their preferences and determine the pace of the class. Among the topics we simply cannot overlook are: random variables, moments, conditional probability, independence, the law of large numbers, the central limit theorem, measure spaces, sigma-algebras, the Lebesgue integral, absolute continuity of measures...
Possible more advanced topics include: the Hausdorff measure for fractal sets, conditional expectations, zero-one laws, and the subjects mentioned above.
Also, an idea could be to use a few lectures to invite people like statisticians, physicists, Wall Street brokers, to speak on how they use probability in their jobs.
In this class, students will be treated as adults and thus responsible for their own learning. That is to say, homework will be kept to a minimal level, and the grading will be done on the basis of the participation in class, e.g., through seminars that the students could give in lieu of the midterm exams.
Textbooks: There are two recommended books for the class:
P. Billingsley, Probability and Measure, Wiley,
1986.
Ya. G. Sinai, Probability: An Introductory Course,
Springer-Verlag, 1992.
There is another very good book. This didn't make it into the recommended list because it might be a little hard, in the beginning. It is a good idea, however, to try and take a look at it:
A. N. Shiryayev (sometimes spelled Shiriaev), Probability, Springer-Verlag, 1984.
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Last updated: Jan 25, 2000 (adapted to ML's Stevens website: Oct 27, 2004)