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Contents

Preface

• Introduction
• Credits
• External Resources

Expository Chapters

0. Foundations

   1. Sets
   2. Functions
   3. Relations
   4. Partial Orders
   5. Equivalence Relations
   6. Cardinality
   7. Counting Measure
   8. Combinatorial Structures
   9. Topological Spaces
   10. Metric Spaces
   11. Measurable Spaces
   12. Measure Spaces
   13. Special Set Structures
   14. Existence and Uniqueness
   15. The Integral With Respect to a Measure
   16. Properties of the Integral
   17. General Measures
   18. Absolute Continuity and Density Functions
   19. Function Spaces
1. Probability Spaces

   1. Random Experiments
   2. Events and Random Variables
   3. Probability Measures
   4. Conditional Probability
   5. Independence
   6. Convergence
   7. Probability Spaces Revisited
   8. Stochastic Processes
   9. Filtrations and Stopping Times
2. Distributions

   1. Introduction
   2. Discrete Distributions
   3. Continuous Distributions
   4. Mixed Distributions
   5. Joint Distributions
   6. Conditional Distributions
   7. Distribution and Quantile Functions
   8. Transformations of Random Variables
   9. Convergence in Distribution
3. Expected Value

   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
   8. Expected Value and Covariance Matrices
   9. Expected Value as an Integral
   10. Conditional Expected Value Revisited
   11. Vector Spaces of Random Variables
   12. Uniformly Integrable Variables
   13. Kernels and Operators
4. Special Distributions

   • Location-Scale Families
   • General Exponential Families
   • Stable Distributions
   • Infinitely Divisible Distributions
   • Power Series Distributions
   • The Normal Distribution
   • The Multivariate Normal Distribution
   • The Gamma Distribution
   • Chi-Square and Related Distribution
   • The Student t Distribution
   • The F Distribution
   • The Lognormal Distribution
   • The Folded Normal Distribution
   • The Rayleigh Distribution
   • The Maxwell Distribution
   • The Lévy Distribution
   • The Beta Distribution
   • The Beta Prime Distribution
   • The Arcsine Distribution
   • General Uniform Distributions
   • The Uniform Distribution on an Interval
   • Discrete Uniform Distributions
   • The Semicircle Distribution
   • The Triangle Distribution
   • The Irwin-Hall Distribution
   • The U-Power Distribution
   • Sine Distributions
   • The Laplace Distribution
   • The Logistic Distribution
   • The Extreme Value Distribution
   • The Hyperbolic Secant Distribution
   • The Cauchy Distribution
   • The Exponential-Logarithmic Distribution
   • The Gompertz Distribution
   • The Log-Logistic Distribution
   • The Pareto Distribution
   • The Wald Distribution
   • The Weibull Distribution
   • Benford's Law
   • The Zeta Distribution
   • The Logarithmic Series Distribution
5. 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
6. Point Estimation

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

   1. Introduction
   2. Estimation the Normal Model
   3. Estimation in the Bernoulli Model
   4. Estimation in the Two-Sample Normal Model
   5. Bayesian Set Estimation
8. 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
9. Geometric Models

   1. Buffon's Problems
   2. Bertrand's Paradox
   3. Random Triangles
10. 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
11. 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
12. Games of Chance

    1. Introduction
    2. Poker
    3. Bridge
    4. Simple Dice Games
    5. Craps
    6. Roulette
    7. The Monty Hall Problem
    8. Lotteries
    9. The Red and Black Game
    10. Timid Play
    11. Bold Play
    12. Optimal Strategies
13. The Poisson Process

    1. Introduction
    2. The Exponential Distribution
    3. The Gamma Distribution
    4. The Poisson Distribution
    5. Thinning and Superpositon
    6. Non-homogeneous Poisson Processes
    7. Compound Poisson Processes
    8. Poisson Processes on General Spaces
14. Renewal Processes

    1. Introduction
    2. Renewal Equations
    3. Renewal Limit Theorems
    4. Delayed Renewal Processes
    5. Alternating Renewal Processes
    6. Renewal Reward Processes
15. Markov Processes

    1. Introduction
    2. Potentials and Generators
    3. Introduction to Discrete Time Chains
    4. Recurrence and Transience
    5. Periodicity
    6. Stationary and Limiting Distributions
    7. Time Reversal in Discrete Time Chains
    8. The Ehrenfest Chains
    9. The Bernoulli-Laplace Chain
    10. Reliability Chains
    11. Discrete Time Branching Chains
    12. Queuing Chains
    13. Discrete Time Birth Death Chains
    14. Random Walks on Graphs
    15. Introduction to Continuous Time Chains
    16. Transition Matrices and Generators
    17. Potential Matrices
    18. Stationary and Limiting Distributions
    19. Time Reversal in Continuous Time Chains
    20. Chains Subordinate to the Poisson Process
    21. Continuous Time Birth-Death Chains
    22. Queuing Chains
    23. Continuous Time Branching Chains
16. Martingales

    1. Introduction
    2. Properties and Constructions
    3. Stopping Times
    4. Inequalities
    5. Convergence
    6. Backwards Martingales
17. Brownian Motion

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

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