# Math 21-325 - Probability theory Sevak Mkrtchyan 7121 Wean Hall e-mail: sevakm (at_sign) andrew.cmu.edu

## Course Description

Concepts covered will include probability spaces, random variables, expectations, conditional probability and independence, limit theorems such as the strong law of large numbers and the central limit theorem, random walks. Additional topics will be covered as time permits.

## Lectures

MWF 9:30-10:20AM in PH 125C

## Office Hours:

MWF 8:30-9:20 and 10:30-11:20 (7121 Wean Hall)

## Textbook

The textbook for the course is: Feller, An Introduction to Probability Theory and Its Applications , Volume 1, 3rd edition

Your grade in the class will be based on the following weights:
25% - Homework assignments
20% - Midterm exam 1 (Monday, October 1, in class)
20% - Midterm exam 2 (Monday, November 5, in class)
35% - Final exam

## Homework

Homework assignments will be posted online and collected in class at the beginning of class on Wednesdays. No late homework will be accepted. The two lowest homework grades will be dropped.

You are allowed to collaborate on homework assignments, however writing up of the solutions should be done individually. You should never share written down solutions which will be submitted as homework by other students.

This is a schedule for what I have covered and what I plan to cover in class each day. This section will be updated regularly during the semester.

 Date Topic Homework Due Mon, August 27 Introduction. The sample space. Chapter 1.8: 4,16; Sep 5 Wed, August 29 Probabilities. Chapter 2.10: 1,3,4,5,11,15. Sep 5 Fri, August 31 Combinatorial Analysis Mon, September 3 Labor day; No classes Wed, September 5 Combinatorial Analysis, Binomial series Chapter 2.10: 17, 19, 27, 32, 38 Sep 12 Fri, September 7 Hypergeometric distribution Chapter 2.11: 3,5,6 Sep 12 Mon, September 10 Waiting times Chapter 2.10: 35,36 Sep 19 Wed, September 12 Random walks Chapter 3.10: 1,2,3,4 Sep 19 Fri, September 14 Random walks - the reflection principle Part of homework 3 Sep 19 Mon, September 17 Random walks - first and last return probabilities Chapter 3.10: 6,7,11,12 Sep 26 Wed, September 19 Random walks - steps on the positive side, maxima Fri, September 21 Combination of events, Conditional probability Chapter 4.6: 3,8,12,15 Sep 26 Mon, September 24 Stratified populations, Independence Chapter 5.8: 2,6,7,15,16, Oct 5 Wed, September 26 Binomial distribution, law of large numbers Chapter 6.10: 3,5,6 Oct 5 Fri, September 28 Poisson distribution, Poisson process Mon, October 1 Midterm 1 Wed, October 3 Normal distribution Chapter 6.10: 11,12,17,20,23,27,41 Oct 17 Fri, October 5 Expectation, variance, standard deviation Part of homework 6 Oct 17 Mon, October 8 More on the normal distribution Wed, October 10 Midterm exam overview Fri, October 12 DeMoivre-Laplace limit theorem Mon, October 15 Examples; Normal approximation to the poisson distribution Wed, October 17 Large deviations Chapter 7.7: 2,4,5,7,9,12,13,14 Oct 26 Fri, October 19 Mid-Semester Break; No Classes Mon, October 22 Infinite Bernoulli trials Wed, October 24 Borel-Cantelli Lemmas Part of homework 8 Nov 2 Fri, October 26 Strong law of large numbers Chapter 8: 1,2,3,4,5,6 Nov 2 Mon, October 29 Conditional distribution and conditional expectation Extra credit problem (Blackboard) Nov 9 Wed, October 31 Joint distribution Part of homework 9 Nov 9 Fri, November 2 Covariance Chapter 9: 2,8,9,10,21,34 Nov 9 Mon, November 5 Midterm 2 Wed, November 7 The weak law of large numbers Chapter 10:1,2ab,6,8 Nov 16 Fri, November 9 Absolutely continuous random variables Part of homework 10 Nov 16 Mon, November 12 Joint density, marginals, conditional expectation Wed, November 14 The partition theorem Fri, November 16 Transformed random variables Homework 11 Nov 30 Mon, November 19 The multivariate normal distribution Wed, November 21 Thanksgiving Holiday; No Classes Fri, November 23 Thanksgiving Holiday; No Classes Mon, November 26 Generating functions Wed, November 28 Characteristic functions Fri, November 30 Proof of the Central Limit Theorem. Brownian Motion Homework 12 Dec 7 Mon, December 3 Brownian Motion Wed, December 5 Brownian Bridge Fri, December 7 Construction of a Poisson process

You are expected to attend every class and arrive on time. It is your responsibility to be informed of any announcements made in class.