Math 605: In this course we will build up the tools and foundations of probability theory needed for understanding and using statistical theories, stochastic simulation, and stochastic processes. Topics covered in this class are measure and integration (construction of probability spaces), distribution functions, random variables and their simulation, convergence of random variables, laws of large numbers and Monte-Carlo methods, concentration inequalities, central limit theorem, information theory, random walks.

A class in real analysis (integration theory, Math 623) is not required for this class as we will review, as needed, the basic constructions of probability spaces. It can be beneficial to take these two classes concurrently for the student who wants and in-depth understanding of the mathematical foundations of probability.

The primary goal of the class is to understand and master the basic probability tools and concepts needed in modern applications such as in stochastic simulation, data science and machine learning, engineering, and economics. Motivated students with various backgrounds are welcome.

The class is the first part of the sequence Math605/606 and Math 606 will cover the theory of stochastic processes and stochastic simulation: Martingale theory, Poisson processes, Markov chains in discrete and continuous time, Markov chain Monte-carlo, Brownian motion.