STAT 25100 Lecture Notes

✍ Scribed by Pierre Yves Gaudreau Lamarre

• Year: 2021
• Tongue: English
• Leaves: 213
• Category: Library

✦ Synopsis

A collection of lecture notes for the undergraduate course "Introduction to Mathematical Probability" offered at the University of Chicago.

✦ Table of Contents

Preface
Chapter 1. Introduction
1.1. What is Probability?
1.2. What is Mathematical Probability?
Chapter 2. The Foundations of Mathematical Probability
2.1. The Sample Space
2.2. Events
2.3. The Probability Measure
Chapter 3. The Uniform Measure
3.1. The Uniform Measure and Counting Problems
3.2. Counting Techniques
3.3. A Return to the Two Examples
3.4. Is the Uniform Measure a Good Model? (Bonus)
Chapter 4. Conditioning and Independence
4.1. Evolution and Interactions in Informal Terms
4.2. Conditional Probability
4.3. Independence
4.4. The Law of Total Probability
4.5. Bayes' Rule
4.6. Two Additional Remarks on Bayes' Rule (Bonus)
Chapter 5. Discrete Random Variables, Expected Value, and Variance
5.1. A Motivating Example
5.2. Discrete Random Variables and Their Distributions
5.3. Expected Value and Variance
5.4. Conditioning and Independence
Chapter 6. Some Important Examples of Discrete Random Variables
6.1. Indicator Random Variables and Processes
6.2. Counting Random Variables and the Binomial Distribution
6.3. First Arrival Times and the Geometric Distribution
6.4. The Poisson Distribution
Chapter 7. The Law of Large Numbers
7.1. The Weak Law of Large Numbers
7.2. A Problem with Theorem 7.1
7.3. Markov's and Chebyshev's Inequalities
7.4. An Example
7.5. Closing Remarks
7.6. The Strong Law of Large Numbers (Bonus)
7.7. Concentration of Measure (Bonus)
Chapter 8. Introduction to Continuous Random Variables
8.1. A Model of Continuous Arrivals
8.2. Uniform Random Number on the Interval [0,1]
8.3. A Solution to the Apparent Paradox
8.4. A Brief Comment on Philosophical Implications
Chapter 9. The Theory of Continuous Random Variables
9.1. Continuous Distributions
9.2. Continuous Expected Values
9.3. Conditioning and Independence with Continuous Variables
9.4. Change of Variables
9.5. The Borel-Kolmogorov Paradox
Chapter 10. The Central Limit Theorem
10.1. The Gaussian Distribution and Universality
10.2. The Central Limit Theorem
10.3. Applications of the Central Limit Theorem
10.4. A Sketch of the Proof
10.5. Accuracy of the Central Limit Theorem (Bonus)
10.6. Alternate Proofs of the Central Limit Theorem (Bonus)