Georgia Institute of Technology College of Sciences

1. Probability Measures: Including connections to distribution functions
2. Random Variables: Including random vectors and discrete-parameter stochastic processes
3. Expectation: Basic properties; convergence theorems; inequalities
4. Independent Random Variables: Basic properties; connections to infinite-dimensional product measures; Fubini's theorem
5. Modes of Convergence of Random Variables: Almost sure convergence; the Borel-Cantelli lemma; convergence in probability; convergence in L^p
6. Laws of Large Numbers: Weak and strong laws; Kolmogorov's inequality; equivalent sequences; random series
7. Convergence in Distribution: Basic properties; connections to sequential compactness, tightness, and uniform integrability
8. Characteristic Functions: Basic properties; connections to probability measures
9. The Classical Central Limit Theorem
10. Conditional Probability and Expectation: Basic properties; connections to Radon-Nikodym derivatives, projections, etc.

Suggested textbooks: A Course in Probability Theory by Chung.
Probability: Theory and Examples by Rick Durrett
Probability and Measure by Patrick Billingsley Probability by Albert Shiryaev
Suggested courses: 6241 and 6242