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Math 170B: General Course Outline

Catalog Description

170B. Probability Theory. (4) Lecture, three hours; discussion, one hour. Requisite: course 170A. Convergence in distribution, normal approximation, laws of large numbers, Poisson processes, random walks. P/NP or letter grading.

Additional Information Probability and stochastic processes are used to create and analyze models in a broad range of fields, including statistics, economics, finance, engineering, biology and physics. Mathematics 170AB and 171 are designed to give a firm foundation in this area for students who will work and/or do graduate work in one of these fields. They also provide an excellent background for graduate work in probability and related areas of mathematics.

These courses are particularly well suited to students who plan to take the exams in actuarial science. The second exam in this series (number 110) is on probability and statistics. Mathematics 170AB covers roughly 2/3 of the material on that exam.

Course 170A is multiply listed with Statistics. Usually, two sections are offered each Fall Quarter, one by Mathematics and one by Statistics. Total enrollment in the two sections tends to be about 50.

The three courses are intended as a year-long sequence. However, it is possible, and not unusual, to take 171 without 170B. In fact, the enrollments in 171 are sometimes larger than in 170B (both are in the 10-20 range). Mathematics 170B is offered each Winter Quarter, and Mathematics 171 is offered each Spring.

Textbook

Introduction to Probability by D. P. Bertsekas and John N. Tsitsiklis, 2nd edition

Comments:

(a) The theoretical problems, which appear with full solutions at the ends of the chapters, are an essential part of the course. Many of them should be incorporated into the lectures. Some to emphasize are given in parentheses below.

(b) Unfortunately, solutions to all of the other problems are freely available on the book?s web site. Additional problems (with no posted solutions) are available at http://www.athenasc.com/prob-supp.html

Outline T. Liggett 4/10

Schedule of Lectures

Lecture Section Topics

1-3

4.1

Derived Distributions

4-5

4.2

Correlation (see comment below)

6-7

4.3

Conditional Expectation and Variance (problem 28)

8-9

4.4

Transforms

10

4.5

Sum of a Random Number of Independent RVs.

11

Midterm Exam

12

5.1

Markov and Chebyshev Inequalities

13

5.2

Weak Law of Large Numbers

14

5.3

Convergence in Probability

15-16

5.4

Central Limit Theorem

17-18

5.5

Strong Law of Large Numbers

19-21

6.1

Bernoulli Process

22

Midterm Exam

23-26

6.2

Poisson Process

27-28

Random Walks or Estimation (see comment below)