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Basic Information

Expository Chapters

Foundations

Sets Functions and Relations

  1. Sets
  2. Functions
  3. Relations
  4. Partial Orders
  5. Equivalence Relations

Cardinality and Combinatorics

  1. Cardinality
  2. Counting Measure
  3. Combinatorial Structures

Topology

  1. Topological Spaces
  2. Metric Spaces

Measure Theory

  1. Measurable Spaces
  2. Special Set Structures

Probability Spaces

Basic Topics

  1. Random Experiments
  2. Events and Random Variables
  3. Probability Measures
  4. Conditional Probability
  5. Independence

Special and Advanced Topics

  1. Convergence
  2. Measure Spaces
  3. Existence and Uniqueness

Distributions

Basic Topics

  1. Discrete Distributions
  2. Continuous Distributions
  3. Mixed Distributions
  4. Joint Distributions
  5. Conditional Distributions
  6. Distribution and Quantile Functions
  7. Transformations of Random Variables

Special and Advanced Topics

  1. Convergence in Distribution
  2. General Distribution Functions
  3. The Integral With Respect to a Measure
  4. Properties of the integral
  5. General Measures
  6. Absolute Continuity and Density Functions
  7. Function Spaces

Expected Value

Basic Topics

  1. Definitions and Basic Properties
  2. Additional Properties
  3. Variance
  4. Skewness and Kurtosis
  5. Covariance and Correlation
  6. Generating Functions
  7. Conditional Expected Value

Special and Advanced Topics

  1. Expected Value and Covariance Matrices
  2. Expected Value as an Integral
  3. Conditional Expected Value Revisited
  4. Vector Spaces of Random Variables
  5. Kernels and Operators

Special Distributions

General Families

Normal

Statistical

Associated with the Normal

Beta and Related

Uniform

Based on Simple Curves

Continuous with support \( \R \)

Continuous with Positive Support

Other Special Models

Random Samples

  1. Introduction
  2. The Sample Mean
  3. The Law of Large Numbers
  4. The Central Limit Theorem
  5. The Sample Variance
  6. Order Statistics
  7. Sample Correlation and Regression
  8. Special Properties of Normal Samples

Point Estimation

  1. Estimators
  2. The Method of Moments
  3. Maximum Likelihood
  4. Bayes Estimators
  5. Best Unbiased Estimators
  6. Sufficient, Complete and Ancillary Statistics

Set Estimation

  1. Introduction
  2. Estimation in the Normal Model
  3. Estimation in the Bernoulli Model
  4. Estimation in the Two-Sample Normal Model
  5. Bayesian Set Estimation

Hypothesis Testing

  1. Introduction
  2. Tests in the Normal Model
  3. Tests in the Bernoulli Model
  4. Tests in the Two-Sample Normal Model
  5. Likelihood Ratio Tests
  6. Chi-Square Tests

Geometric Models

  1. Buffon's Problems
  2. Bertrand's Paradox
  3. Random Triangles

Bernoulli Trials

  1. Introduction
  2. The Binomial Distribution
  3. The Geometric Distribution
  4. The Negative Binomial Distribution
  5. The Multinomial Distribution
  6. The Simple Random Walk
  7. The Beta-Bernoulli Process

Finite Sampling Models

  1. Introduction
  2. The Hypergeometric Distribution
  3. The Multivariate Hypergeometric Distribution
  4. Order Statistics
  5. The Matching Problem
  6. The Birthday Problem
  7. The Coupon Collector Problem
  8. Pólya's Urn Process
  9. The Secretary Problem

Games of Chance

Basic Topics

  1. Introduction
  2. Poker
  3. Simple Dice Games
  4. Craps
  5. Roulette
  6. The Monty Hall Problem
  7. Lotteries

Red and Black

  1. The Game of Red and Black
  2. Timid Play
  3. Bold Play
  4. Optimal Strategies

The Poisson Process

Basic Topics

  1. Introduction
  2. The Exponential Distribution
  3. The Gamma Distribution
  4. The Poisson Distribution

Variations

  1. Thinning and Superposition
  2. Non-homogeneous Poisson Processes
  3. Compound Poisson Processes
  4. Poisson Processes on General Spaces

Renewal Processes

Basic Topics

  1. Introduction
  2. Renewal Equations
  3. Renewal Limit Theorems

Variations and Applications

  1. Delayed Renewal Processes
  2. Alternating Renewal Processes
  3. Renewal Reward Processes

Stochastic Processes

Basic Topics

  1. Introduction
  2. Filtrations and Stopping Times

Special Classes of Processes

  1. Gaussian Processes
  2. Processes with Stationary, Independent Increments
  3. Martingales

Markov Processes

General Theory

  1. Introduction
  2. Potentials and Generators

Discrete-Time Markov Chains

  1. Discrete-Time Chains
  2. Recurrence and Transience
  3. Periodicity
  4. Invariant and Limiting Distributions
  5. Time Reversal

Special Discrete-Time Chains

  1. The Ehrenfest Chains
  2. The Bernoulli-Laplace Chain
  3. Reliability Chains
  4. The Branching Chain
  5. Queuing Chains
  6. Birth-Death Chains
  7. Random Walks on Graphs

Continuous-Time Markov Chains

  1. Basic Structure
  2. Transition Matrices and Generators
  3. Potential Matrices

Brownian Motion

Basic Topics

  1. Standard Brownian Motion

Variations

  1. Brownian Motion with Drift and Scaling
  2. The Brownian Bridge Process
  3. Geometric Brownian Motion

Ancillary Materials

Apps

Simple Probability and Combinatorics

Geometric Models

Bernoulli Trials

Finite Sampling Models

Dice

Games of Chance

Renewal Processes

Interacting Particle Systems

Special Distributions

Random Samples

Point Estimation

Interval Estimation

Hypothesis Testing

Markov Processes

Brownian Motion

Data Sets

Biographies

Art

Paintings