**Probability Theory I**

M170A, University of California, Los Angeles, Winter 2022

**Abstract:**The course discusses the rigorous foundations of probability theory based on real analysis. It starts with the axiomatic definition of probability, and it covers 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, and central limit theorem.

**Textbook:**G. R. Grimmett and D. J. Welsh.

*Probability: An Introduction*, (2nd edition, 2014)

**TA:**Zehan Chao

**Homework and exams:**

Homework #1 (01/15)

Homework #2 (01/22)

Homework #3 (01/29)

Midterm (02/07)

Homework #4 (02/12)

Homework #5 (02/19)

Homework #6 (02/26)

Homework #7 (03/05)

Homework #8 (not graded)

Final exam (03/17)

**Table of contents:**

I

*Formalism of probability theory*

1 Introduction

2 Outcomes, events

3 Probabilities

4 Continuity of probabilities

5 Conditional probabilities

6 Independence

7 Partition theorem

8 Borel-Cantelli lemmas

II

*Discrete random variables*

1 Probability mass function

2 Examples of discrete random variables

3 Functions of random variables

4 Expectation and variance

5 Conditional expectation

III

*Multivariate discrete distributions*

1 Joint distributions

2 Independence

3 Inclusion-exclusion formula

IV

*Continuous random variables*

1 Cumulative distribution function

2 Continuity vs. absolute continuity

3 Expectation, median, and variance

4 Functions of random variables, change of variables

5 Examples of continuous random variables

6 Poisson process

7 Geometric probability

V

*Multivariate continuous distributions*

1 Joint distributions, densities, and independence

2 Sums of random variables

3 Functions of random variables, change of variables

4 Expectations of functions of random variables

5 Covariance, correlations

6 Least squares

7 Cauchy-Schwarz and Hölder inequalities

8 Continuous conditional distributions

9 Bivariate normal distribution

VI

*Some useful tools*

1 Moment generating function

2 Characteristic function

3 Inequalities on tail probabilities (Markov, Chebychev, Cantelli, Paley-Zygmund)

4 Kocher-Stone's Borel-Cantelli lemma

VII

*Main limit theorems*

1 Notions of convergence for random variables

2 Laws of large numbers

3 Convergence in distribution

4 Central limit theorem

5 Confidence intervals

6 Concentration inequalities (Hoeffding, Bernstein)

7 Large deviation principle

**Schedule:**

Course #1 (01/03, 12pm-1pm): Sections I.1-I.3

Course #2 (01/05, 12pm-1pm): Sections I.3-I.4

Course #3 (01/07, 12pm-1pm): Sections I.5-I.7

Course #4 (01/10, 12pm-1pm): Section I.7-I.8

Course #5 (01/12, 12pm-1pm): Section I.8

Course #6 (01/14, 12pm-1pm): Section II.1-II.2

Course #7 (01/17, 12pm-1pm): Section II.2

Course #8 (01/19, 12pm-1pm): Section II.3-II.4

Course #9 (01/21, 12pm-1pm): Section II.4-II.5

Course #10 (01/24, 12pm-1pm): Section III.1-III.3

Course #11 (01/26, 12pm-1pm): Section III.3-IV.1

Course #12 (01/28, 12pm-1pm): Section IV.2-IV.4

Course #13 (01/31, 12pm-1pm): Section IV.4-IV.5

Course #14 (02/02, 12pm-1pm): Section IV.5

Course #15 (02/04, 12pm-1pm): Section IV.6-IV.7

Course #16 (02/09, 12pm-1pm): Section V.1-V.1

Course #17 (02/11, 12pm-1pm): Section V.3-V.5

Course #18 (02/14, 12pm-1pm): Section V.6-V.7

Course #19 (02/16, 12pm-1pm): Section V.8-V.9

Course #20 (02/18, 12pm-1pm): Section V.9

Course #21 (02/21, 12pm-1pm): Section VI.1-VI.2

Course #22 (02/23, 12pm-1pm): Section VI.3

Course #23 (02/25, 12pm-1pm): Section VI.3-VI.4

Course #24 (02/28, 12pm-1pm): Section VII.1-VII.2

Course #25 (03/02, 12pm-1pm): Section VII.3-VII.4

Course #26 (03/04, 12pm-1pm): Section VII.5-VII.6

Course #27 (03/07, 12pm-1pm): Section VII.6-VII.7