Difference between revisions of "Math 636 Advanced Probability 1"
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Contents
Catalog Information
Title
Advanced Probability 1.
Credit Hours
3
Prerequisite
Math 314 and Math 341; and Math 431 or Stat 370; or equivalents.
Description
Foundations of the modern theory of probability with applications. Probability spaces, random variables, independence, conditioning, expectation, generating functions, and Markov chains.
Desired Learning Outcomes
This should be an advanced course in probability and, therefore, clearly distinguishable from an introductory course like Math 431. Furthermore, it is supposed to be a course in the modern theory of probability, which suggests that it should be based on Kolmogorov's measuretheoretic approach or something equivalent.
Prerequisites
The official prerequisite is multivariable calculus. Other prior courses that will contribute to student success include:
 an introductory course in probability;
 a course in rigorous mathematical reasoning;
 an introductory course in analysis.
Minimal learning outcomes
Outlined below are topics that all successful Math 543 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.
 Probability spaces
 Sigmaalgebras and Borel sets
 Kolmogorov axioms
 Carathéodory's Extension Theorem
 LebesgueStieltjes measure
 Random variables
 Measurable maps
 Distributions and distribution functions
 Independence
 Of events and classes of events
 Of random variables
 BorelCantelli Lemmas
 Expectation
 Of arbitrary nonnegative random variables
 Of integrable realvalued random variables
 Of compositions
 Monotone Convergence Theorem
 Uniform integrability and dominated convergence
 Conditioning
 Probability conditioned on a nonnull set
 Expectation conditioned on a sigmaalgebra
 Expectation conditioned on a random variable
 Probability measures on product spaces
 Strong Law of Large Numbers
 Central Limit Theorem
 Convergence of random variables
 Almost sure
 In probability
 L^p
 weak
 Discretetime Martingales
Textbooks
Possible textbooks for this course include (but are not limited to):
 Achim Klenke, Probability Theory: A Comprehensive Course, Springer, 2008.