# Math 170A: General Course Outline

## Catalog Description

**170A. Probability Theory. (4) **

Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33A, 131A. Not open to students with credit in course 170E, Electrical Engineering 131A or Statistics 100A. Rigorous presentation of probability theory based on real analysis. Probability space, probability and conditional probability, independence, Bayes? rule, discrete and continuous random variables and their distributions, expectation, moments and variance, conditional distribution and expectation, weak law of large numbers. P/NP or letter grading.

## Course Information:

The course discusses the foundations of probability as a mathematical discipline rooted in undergraduate real analysis. At the end of the course, the students will have the tools and ability to formulate, analyze an answer questions in probability and prove the validity of their reasoning in full mathematical rigor.

**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

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

## Schedule of Lectures

Lecture | Section | Topics |
---|---|---|

Week 1 |
Sections 1.1-1.5, 1.9 |
Sample space, events, probability |

Week 2 |
Sections 1.6-1.7 |
Conditional probability and independence |

Week 3 |
Sections 1.8, 1.10 |
Partition theorem and Bayes rule, examples |

Week 4 |
Sections 2.1-2.4 |
Discrete random variables, their functions, expectation and variance |

Week 5 |
Sections 2.5, 3.1-3.2 |
Conditional expectation, multivariate discrete distributions |

Week 6 |
Sections 3.3-3.5 |
Independent discrete random variables, indicators |

Week 7 |
Sections 5.1-5.6 |
Cumulative distribution function, continuous random variables |

Week 8 |
Sections 6.1-6.4 |
Multivariate distributions and their marginals |

Week 9 |
Sections 6.5-6.7 |
Change of variables, conditional expectation |

Week 10 |
Sections 6.8, 7.3, 8.1-8.2 |
Mutlivariate normal distribution, weak law of large numbers |