brani@stat.duke.edu
This is a course about random variables, their convergence and conditional expectations, motivating an introduction to the foundations of statistical inference. It is a course for statisticians, and does not give detailed coverage to abstract measure theory or the abstract mathematics of probability theory. For this you should consider MTH 241 or MTH 287.
The course synopsis reads: Introduction to probability spaces, the theory of measure and integration, random variables, and limit theorems. Distribution functions, densities, and characteristic functions; convergence of random variables and of their distributions; uniform integrability and the Lebesgue convergence theorems. Weak and strong laws of large numbers, central limit theorem.
Mathematical topics from real analysis, including parts of measure theory, Fourier and functional analysis, are introduced as needed to support understanding of axiomatic probability and its applications.
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| Brani Vidakovic |
|---|---|
| Office: 223 B Old Chemistry Building | |
| Office Hours: Tuesdays 2:15-3:30; Thursdays 2:15-3:30 | |
| Phone: 684-8025 | |
| Email: brani@stat.duke.edu |
Tentative Outline:
Mathematical foundations of probability
theory (Kolmogorov's axioms, Algebras and sigma-algebras, Measirable spaces,
Random variables, Conditional probabilities, Conditional expectations,
Various kinds of convergence of random variables)
Hilbert space of random variables with
finite second moments.
Characteristic functions.
Convergence of probability measures
(Tightness, Relative compactness, Central limit theorem,
Lindenberg and some nonclassical conditions for CLT)
Sequences and sums of random variables
(Zero-one laws, Convergence of series, Strong law of large numbers,
Law of itterated logarithm
Martingales; Fundamental inequalities; Convergence results.
Please send comments to
brani@stat.duke.edu