Part I: Foundations

Foundations

• Chapter 1: Probability Basics
  • 1.1 Sample Spaces and Events
  • 1.2 The Kolmogorov Axioms
  • 1.3 Conditional Probability
  • 1.4 The Law of Total Probability
  • 1.5 Bayes’ Theorem
  • 1.6 Independence
  • 1.7 Summary
• Chapter 2: Random Variables
  • 2.1 What Is a Random Variable?
  • 2.2 Probability Mass Function (PMF)
  • 2.3 Probability Density Function (PDF)
  • 2.4 Cumulative Distribution Function (CDF)
  • 2.5 Expectation (Expected Value)
  • 2.6 Variance
  • 2.7 Moment Generating Functions
  • 2.8 Joint Distributions
  • 2.9 Covariance and Correlation
  • 2.10 Conditional Distributions and Conditional Expectation
  • 2.11 The Law of Large Numbers
  • 2.12 The Central Limit Theorem
  • 2.13 Summary
• Chapter 3: Common Distributions
  • 3.1 Discrete Distributions
  • 3.2 Continuous Distributions
  • 3.3 Relationships Between Distributions
  • 3.4 Connecting the Hospital Story
  • 3.5 Summary
• Chapter 4: The Likelihood Function
  • 4.1 Likelihood vs. Probability: The Key Distinction
  • 4.2 Formal Definition of the Likelihood Function
  • 4.3 The Log-Likelihood
  • 4.4 Finding the MLE Analytically
  • 4.5 The Score Function
  • 4.6 Fisher Information
  • 4.7 Sufficient Statistics
  • 4.8 The Likelihood Principle
  • 4.9 Bringing It All Together: Complete Light Bulb Analysis
  • 4.10 Summary