This is a course about random variables, especially about their convergence and conditional expectations, providing an introduction to the foundations of modern Bayesian statistical inference.
Students are expected to know real analysis at the level of W. Rudin's Principles of Mathematical Analysis or M. Reed's Fundamental Ideas of Analysis--- the topology of R^n, convergence in metric spaces (especially uniform convergence of functions on R^n), infinite series, countable and uncountable sets, compactness and convexity, and so forth. Students without this background should take or at least co-register with Duke's Math 203, Basic Analysis I. More advanced mathematical topics from real analysis, including parts of measure theory, Fourier and functional analysis, are introduced as needed to support a deep understanding of probability and its applications. Topics of later interest in statistics (e.g., regular conditional density functions) are given special emphasis. |
