Here are 100 chapter titles for a book on Probability Theory, progressing from beginner to advanced:
I. Foundations and Basic Concepts (20 Chapters)
- Introduction to Probability: What is Randomness?
- Sample Spaces and Events
- The Axioms of Probability
- Basic Probability Calculations: Addition Rule, Complement Rule
- Conditional Probability: Definition and Examples
- The Multiplication Rule and Bayes' Theorem
- Independence of Events
- Combinatorial Probability: Counting Techniques (Permutations, Combinations)
- Discrete Random Variables: Definition and Examples
- Probability Mass Functions (PMFs)
- Common Discrete Distributions: Bernoulli, Binomial, Poisson
- Expectation of Discrete Random Variables
- Variance and Standard Deviation
- Joint Distributions: Discrete Case
- Conditional Distributions: Discrete Case
- Independence of Random Variables
- Functions of Random Variables
- Introduction to Stochastic Processes
- Applications of Probability: Basic Examples
- Probability and Statistics: A First Look
II. Continuous Probability and Distributions (30 Chapters)
- Continuous Random Variables: Definition and Examples
- Probability Density Functions (PDFs)
- Cumulative Distribution Functions (CDFs)
- Common Continuous Distributions: Uniform, Exponential, Normal
- Expectation and Variance of Continuous Random Variables
- Joint Distributions: Continuous Case
- Conditional Distributions: Continuous Case
- Functions of Continuous Random Variables
- Transformations of Random Variables
- Order Statistics
- Moment Generating Functions (MGFs)
- Characteristic Functions
- The Central Limit Theorem: Introduction
- Normal Approximation to the Binomial Distribution
- The Law of Large Numbers: Weak and Strong Forms
- Convergence in Probability: Almost Sure, In Probability, Weak
- Convergence in Distribution
- The Delta Method
- Simulation of Random Variables
- Monte Carlo Methods: Introduction
- Applications of Continuous Probability: Examples
- Bivariate Normal Distribution
- Multivariate Normal Distribution
- Exponential Family of Distributions
- Gamma Distribution and its Properties
- Chi-Square Distribution and its Properties
- t-Distribution and F-Distribution
- Weibull Distribution and its Applications
- Extreme Value Theory: Introduction
- Point Processes: Introduction
III. Advanced Probability Theory (30 Chapters)
- Measure Theory: Introduction
- σ-algebras and Measurable Spaces
- Probability Measures and Probability Spaces
- Lebesgue-Stieltjes Integration
- Expectation as a Lebesgue Integral
- Conditional Expectation: Formal Definition
- Martingales: Introduction
- Stopping Times
- Martingale Convergence Theorems
- Brownian Motion: Definition and Properties
- Brownian Motion: Construction
- Brownian Motion: Applications
- Stochastic Calculus: Introduction
- Itô's Lemma
- Stochastic Differential Equations (SDEs): Introduction
- Applications of SDEs: Finance, Physics
- Markov Chains: Discrete Time
- Transition Matrices and Stationary Distributions
- Ergodic Theorems for Markov Chains
- Markov Chains: Continuous Time
- Renewal Theory
- Queueing Theory: Basic Models
- Queueing Theory: Advanced Topics
- Random Walks
- Large Deviations Theory: Introduction
- Concentration Inequalities
- Empirical Processes
- Information Theory: Basic Concepts
- Entropy and Mutual Information
- Applications of Information Theory
IV. Further Explorations and Specialized Topics (20 Chapters)
- Probability and Statistics: Advanced Topics
- Hypothesis Testing
- Estimation Theory
- Bayesian Statistics
- Time Series Analysis
- Spatial Statistics
- Stochastic Geometry
- Random Graphs
- Probabilistic Analysis of Algorithms
- Random Matrix Theory
- Probability in High Dimensions
- Nonparametric Statistics
- Simulation Methods: Advanced Topics
- Markov Chain Monte Carlo (MCMC)
- Applications of Probability in Machine Learning
- Applications of Probability in Deep Learning
- History of Probability Theory
- Philosophical Foundations of Probability
- Open Problems in Probability Theory
- Appendix: Foundational Material and References