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Math 170B: General Course Outline

Catalog Description

170B. Probability Theory II. (4) Lecture, three hours; discussion, one hour. Enforced requisite: courses 170A, 131A. Continuation of rigorous presentation of probability theory based on real analysis. Moments and generating functions; laws of large numbers, the central limit theorem, and convergence in distribution; branching processes; random walks; Poisson and other random processes in continuous time. Advance topics in probability theory. P/NP or letter grading.

Additional Information Advanced Topics:
Instructor selection of three more advanced topics. Possibilities include, but are not limited to:

(1) order statistics, extreme values, and Poisson processes;

(2) basics of entropy and information theory;

(3) theory of sampling and confidence in statistics;

(4) the probabilistic method in combinatorics, including lower bounds for Ramsey numbers;

(5) Borel-Cantelli lemmas, the strong law of large numbers, and Borel's normal number theorem.

Instructor will provide notes or reference materials.

Textbook

Probability: An Introduction (2nd ed.). Oxford: Oxford University Press. by Grimmett, G. R., & Welsh, D. J. (2014).

Schedule of Lectures

Lecture Section Topics

1-3

Review from 170A: probability spaces, random variables, and distributions; multi-variate distributions and independence; discrete and continuous conditional probability

4-6

4.2--4.4, 7.1--7.4, 7.6

Moments, probability and moment generating functions, characteristic functions

7-9

8.1--8.3, 8.5

Inequalities. Laws of large numbers. The central limit theorem. Convergence in distribution.

10-12

9.1--9.5

Branching processes and the method of generating functions, probability of extinction

13-15

10.1--10.4

Random walks on the integers: recurrence vs transience, gambler's ruin

16-18

11.1--11.4

Poisson processes and their inter-arrival times. Population growth, birth processes

19-21

11.5, 11.6

Birth-and-death processes, queueing models

22-24

*Advanced Topic 1

25-27

*Advanced Topic 2

28-30

*Advanced Topic 3