Syllabus
Time and Location
M-W-F Period 5 (11:45 AM – 12:35 PM), LIT 223
Office Hours
Monday 1:55 PM – 2:45 PM, Wednesday 1:55 PM – 2:45 PM, or by appointment
Textbook
There is no required textbook, but the following textbooks are suggested:
- R. Durrett, Probability: Theory and Examples, 5th edition (PDF available on Prof. Durrett’s website)
- D. Khoshnevisan, Probability, Graduate studies in mathematics vol. 80, 2007
Final Exam Date
12/10/2019 @ 12:30 PM – 2:30 PM
Scope of the course
The aim of the course is to provide students with strong foundations in the area of probability theory. At the end of the course, students will be acquainted with the language of probability and will gain sufficient experience to successfully apply probabilistic tools to most areas of pure and applied sciences.
The course is intended for graduate students as part of their PhD requirement, and for students considering studying probability theory at a research level.
Prerequisite
MAA 5228 and MAA 5229
Topics Covered
Topics include Basics of probability theory, Random variables, Independence, Characteristic function, Modes of convergence, Laws of Large Numbers, Central Limit Theorem. Below is the weekly schedule:
W1: Basics of probability theory (probability space, construction of Lebesgue integral).
W2: Random variable, distribution of random variable, transfer lemma.
W3: Basic discrete and continuous distributions, moment generating function.
W4: Functions of random variables.
W5: Conditional probability and independence.
W6: Modes of Convergence: almost sure convergence, convergence in probability, convergence in Lp.
W7: Convergence in distribution, relationships between modes of convergence.
W8: Tightness, Helly and Prohorov theorems.
W9: Law of Large Numbers: weak LLN, strong LLN.
W10: Central Limit Theorem: characteristic function.
W11: Levy’s continuity theorem.
W12: Lyapunov and Linderberg conditions.
W13: Berry-Esseen CLT, confidence interval.
W14: Simulation.