Probability, Mathematical Statistics, Stochastic Processes



Expository Chapters

  1. 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
  2. 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
  3. 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
  4. 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
  5. Special Distributions
  6. 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
  7. 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
  8. 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
  9. 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
  10. Geometric Models
    1. Buffon's Problems
    2. Bertrand's Paradox
    3. Random Triangles
  11. 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
  12. 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
  13. 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
  14. 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
  15. Renewal Processes
    1. Introduction
    2. Renewal Equations
    3. Renewal Limit Theorems
    4. Delayed Renewal Processes
    5. Alternating Renewal Processes
    6. Renewal Reward Processes
  16. 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
  17. Martingales
    1. Introduction
    2. Properties and Constructions
    3. Stopping Times
    4. Inequalities
    5. Convergence
    6. Backwards Martingales
  18. Brownian Motion
    1. Standard Brownian Motion
    2. Brownian Motion with Drift and Scaling
    3. The Brownian Bridge
    4. Geometric Brownian Motion

Ancillary Materials


Random is a website devoted to probability, mathematical statistics, and stochastic processes, and is intended for teachers and students of these subjects. The site consists of an integrated set of components that includes expository text, interactive web apps, data sets, and biographical sketches. Please read the introduction for more information about the content, structure, mathematical prerequisites, technologies, and organization of the project.

Technologies and Browser Requirements

This site uses a number of open and standard technologies, including HTML5, CSS, and JavaScript. To use this project properly, you will need a modern browser that supports these technologies. Display of mathematical notation is handled by the open source MathJax project.

Support and Partnerships

This project was partially supported by a two grants from the Course and Curriculum Development Program of the National Science Foundation (award numbers DUE-9652870 and DUE-0089377). This project was also partially supported by the University of Alabama in Huntsville. Please see the credits for additional information.

Rights and Permissions

This work is licensed under a Creative Commons License. Basically, you are free to copy, distribute, and display this work, to make derivative works, and to make commercial use of the work. However you must give proper attribution and provide a link to the home site: http://www.randomservices.org/random/. Click on the Creative Commons link above for more information.


Kyle Siegrist
Department of Mathematical Sciences
University of Alabama in Huntsville


Mathematics ... is indispensable as an intellectual technique. In many subjects, to think at all is to think like a mathematician.Robert M. Hutchins, The Learning Society