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Department:

MATH

Course Number:

6221

Hours - Lecture:

3

Hours - Lab:

0

Hours - Recitation:

0

Hours - Total Credit:

3

Typical Scheduling:

Fall starting Fall 21

Classical introduction to probability theory including expectation, notions of convergence, laws of large numbers, independence, large deviations, conditional expectation, martingales, and Markov chains.

The three course series MATH 6579, 6580, and 6221 is designed to provide a high level mathematical background for engineers and scientists.

Note that MATH 6221 is not equivalent to MATH 6421, and does not provide any credit towards completion of that course.

Prerequisites:

MATH 6579 or consent of the school.

Course Text:

At the level of Grimmett and Stirzaker, *Probability and Random Processes*

Topic Outline:

- Distribution functions and random variables definition and examples of discrete and continuous distribution functions, discrete and continuous random variables, independence
- Expectation and mode of convergence, expectation and conditional expectation; Markov, Chebychev, Holder, Minkowski and other inequalities, various notions of convergence
- Laws of large numbers and convergence of series, Borel-Cantelli lemmas, Kolmogorov three series theorem, Kolmogorov's strong law
- Large deviations, elements of large deviations, the theorems of Cramer, Hoeffding, Chernoff
- Characteristic functions, the central limit theorem and its rate of convergence (Berry-Esseen inequality)
- Conditional expectations and discrete time, martingales definition and examples of martingales (super and sub), the martingale convergence theorem, L2 bounded and uniformly integrable martingales
- If time permits, Markov chains definitions and examples of Markov chains, invariant measure, rate of convergence, transience and recurrence