Difference between revisions of "Math 543: Advanced Probability 1"
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=== Prerequisite === | === Prerequisite === | ||
| − | [[Math 314]] | + | [[Math 314]] and [[Math 341]]; and [[Math 431]] or Stat 370; or equivalents. |
| − | + | ||
| − | + | ||
| − | [[Math 341]] | + | |
=== Description === | === Description === | ||
| Line 32: | Line 29: | ||
#* Sigma-algebras and Borel sets | #* Sigma-algebras and Borel sets | ||
#* Kolmogorov axioms | #* Kolmogorov axioms | ||
| − | #* Carathéodory's | + | #* Carathéodory's Extension Theorem |
#* Lebesgue-Stieltjes measure | #* Lebesgue-Stieltjes measure | ||
# Random variables | # Random variables | ||
| Line 40: | Line 37: | ||
#* Of events and classes of events | #* Of events and classes of events | ||
#* Of random variables | #* Of random variables | ||
| − | #* | + | #* Borel-Cantelli Lemmas |
# Expectation | # Expectation | ||
#* Of arbitrary nonnegative random variables | #* Of arbitrary nonnegative random variables | ||
#* Of integrable real-valued random variables | #* Of integrable real-valued random variables | ||
#* Of compositions | #* Of compositions | ||
| − | #* Monotone | + | #* Monotone Convergence Theorem |
#* Uniform integrability and dominated convergence | #* Uniform integrability and dominated convergence | ||
# Conditioning | # Conditioning | ||
| Line 51: | Line 48: | ||
#* Expectation conditioned on a sigma-algebra | #* Expectation conditioned on a sigma-algebra | ||
#* Expectation conditioned on a random variable | #* Expectation conditioned on a random variable | ||
| − | #* Bayes' | + | #* Bayes' Formula |
#* Regular conditional distributions | #* Regular conditional distributions | ||
| + | # Probability measures on product spaces | ||
| + | #* Product measures | ||
| + | #* Kolmogorov Extension Theorem | ||
# Generating functions | # Generating functions | ||
| − | # Discrete Markov chains | + | # Discrete Markov chains<br><br><br><br><br><br><br><br> |
</div> | </div> | ||
| + | === Textbooks === | ||
| + | Possible textbooks for this course include (but are not limited to): | ||
| + | |||
| + | * Achim Klenke, ''Probability Theory: A Comprehensive Course'', Springer, 2008. | ||
=== Additional topics === | === Additional topics === | ||
=== Courses for which this course is prerequisite === | === Courses for which this course is prerequisite === | ||
| − | + | [[Math 544]] | |
| − | + | ||
| − | + | ||
Latest revision as of 16:15, 29 March 2018
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 measure-theoretic 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
- Sigma-algebras and Borel sets
- Kolmogorov axioms
- Carathéodory's Extension Theorem
- Lebesgue-Stieltjes measure
- Random variables
- Measurable maps
- Distributions and distribution functions
- Independence
- Of events and classes of events
- Of random variables
- Borel-Cantelli Lemmas
- Expectation
- Of arbitrary nonnegative random variables
- Of integrable real-valued random variables
- Of compositions
- Monotone Convergence Theorem
- Uniform integrability and dominated convergence
- Conditioning
- Probability conditioned on a non-null set
- Expectation conditioned on a sigma-algebra
- Expectation conditioned on a random variable
- Bayes' Formula
- Regular conditional distributions
- Probability measures on product spaces
- Product measures
- Kolmogorov Extension Theorem
- Generating functions
- Discrete Markov chains
Textbooks
Possible textbooks for this course include (but are not limited to):
- Achim Klenke, Probability Theory: A Comprehensive Course, Springer, 2008.
