Math 170B: General Course Outline
Course 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) BorelCantelli lemmas, the strong law of large numbers, and Borel's normal number theorem.
Instructor will provide notes or reference materials.
Textbook(s)
Probability: An Introduction (2nd ed.). Oxford: Oxford University Press. by Grimmett, G. R., & Welsh, D. J. (2014).
Schedule of Lectures
Lecture  Section  Topics 

13 
Review from 170A: probability spaces, random variables, and distributions; multivariate distributions and independence; discrete and continuous conditional probability 

46 
4.24.4, 7.17.4, 7.6 
Moments, probability and moment generating functions, characteristic functions 
79 
8.18.3, 8.5 
Inequalities. Laws of large numbers. The central limit theorem. Convergence in distribution. 
1012 
9.19.5 
Branching processes and the method of generating functions, probability of extinction 
1315 
10.110.4 
Random walks on the integers: recurrence vs transience, gambler's ruin 
1618 
11.111.4 
Poisson processes and their interarrival times. Population growth, birth processes 
1921 
11.5, 11.6 
Birthanddeath processes, queueing models 
2224 
*Advanced Topic 1 

2527 
*Advanced Topic 2 

2830 
*Advanced Topic 3 