fami Topics in Probability Theory
link to University of Nebraska Lincoln
Department of Mathematics
203 Avery Hall
University of Nebraska Lincoln
Lincoln, NE 68588-0323
402-472-3731 (voice)
402-472-8466 (fax)

Topics in
Probability Theory and Stochastic Processes
Steven R. Dunbar


Lessons

Note: In all notes, lessons, tests, quizzes and homework assignments, unless specifically noted otherwise

  1. The lessons use MathML, use a modern browser to view.
  2. The mathematical function log(x) means the natural logarithm, that is, logarithms to the base e.

Real Analysis Background

  1. Orders of Growth
    Updated Friday, 26-Jul-2019 08:50:18 CDT
  2. PDF Version: Orders of Growth
    Updated Friday, 26-Jul-2019 08:50:18 CDT
  1. Asymptotic Limits and Series
    Updated Monday, 03-Oct-2011 06:02:34 CDT
  2. PDF Version: Asymptotic Limits and Series
    Updated Monday, 03-Oct-2011 06:02:27 CDT

Stirling's Formula

  1. A Survey of Stirling's Formula
    Note: PDF file
    Updated Monday, 23-Apr-2012 05:32:53 CDT
  2. Wallis Formula (HTML with Mathjax)
    Updated Friday, 03-Nov-2017 05:24:16 CDT
  3. Wallis Formula (PDF) Wallis Formula
    Updated Thursday, 02-Nov-2017 08:12:13 CDT
  4. Evaluation of the Gaussian Density Integral
    Updated Saturday, 22-Oct-2011 05:49:19 CDT
  5. PDF Version: Evaluation of the Gaussian Density Integral
    Updated Saturday, 22-Oct-2011 05:49:13 CDT
  6. (MathJax Version) Stirling's Formula Derived from Elementary Sequences and Series
    Updated Friday, 26-Jul-2019 08:52:02 CDT
  7. PDF Version: Stirling's Formula Derived from Elementary Sequences and Series
    Updated Friday, 26-Jul-2019 08:52:05 CDT
  8. Stirling's Formula by Using Euler-Maclaurin Summation
    Updated Monday, 10-Oct-2011 06:00:47 CDT
  9. PDF Version: Stirling's Formula by Using Euler-Maclaurin Summation
    Updated Monday, 10-Oct-2011 06:00:40 CDT
  10. Stirling's Formula from Wallis' Formula and the Trapezoidal Approximation
    Updated Monday, 10-Oct-2011 06:00:00 CDT
  11. PDF Version: Stirling's Formula from Wallis' Formula and the Trapezoidal Approximation
    Updated Monday, 10-Oct-2011 05:59:56 CDT
  12. Stirling's Formula from Sums and Taylor Series Approximation
    Updated Friday, 16-Dec-2011 07:39:51 CST
  13. PDF Version: Stirling's Formula from Sums and Taylor Series Approximation
    Updated Friday, 16-Dec-2011 07:39:44 CST
  14. Stirling's Formula Derived from the Gamma Function
    Updated Monday, 26-Nov-2012 05:56:59 CST
    PDF Version: Stirling's Formula Derived from the Gamma Function
    Updated Monday, 26-Nov-2012 05:57:13 CST
  15. Stirling's Formula in Real and Complex Variables
    Updated Monday, 23-May-2011 05:55:15 CDT
    PDF Version: Stirling's Formula in Real and Complex Variables
    Updated Monday, 23-May-2011 05:55:25 CDT
  16. Stirling's Formula Derived from the Poisson Distribution
    Updated [an error occurred while processing this directive]
    PDF Version: Stirling's Formula Derived from the Poisson Distribution
    Updated Friday, 23-Jun-2017 06:01:05 CDT

Bernoulli Trials

  1. Binomial Distribution
    Updated Tuesday, 11-Sep-2018 05:34:57 CDT
  2. PDF Version: Binomial Distribution
    Updated Tuesday, 11-Sep-2018 05:35:03 CDT
  3. Weak Law of Large Numbers
    Updated Wednesday, 25-May-2011 06:16:28 CDT
  4. PDF Version: Weak Law of Large Numbers
    Updated Wednesday, 25-May-2011 06:16:34 CDT
  5. Large Deviations
    Updated Friday, 09-Nov-2018 05:45:27 CST
  6. PDF Version: Large Deviations
    Updated Friday, 09-Nov-2018 05:45:31 CST
  7. de Moivre Laplace Central Limit Theorem
    Updated Friday, 09-Dec-2011 21:16:09 CST
  8. PDF Version: de Moivre Laplace Central Limit Theorem
    Updated Friday, 09-Dec-2011 21:15:59 CST
  9. The Moderate Deviations Result
    Updated Thursday, 29-Nov-2012 05:51:24 CST
  10. PDF Version: The Moderate Deviations Result
    Updated Thursday, 29-Nov-2012 05:51:35 CST
  11. MathJax version The Local Limit Theorem
    Updated Sunday, 26-Apr-2015 09:32:17 CDT
  12. PDF Version: The Local Limit Theorem
    Updated Sunday, 26-Apr-2015 09:32:08 CDT
  13. PDF Version: Ruin Probabilities
    Updated Thursday, 28-Sep-2017 08:40:58 CDT
  14. MathJax Version: Ruin Probabilities
    Updated Thursday, 28-Sep-2017 08:40:58 CDT
  15. PDF Version: Duration of Game
    Updated Thursday, 28-Sep-2017 08:41:18 CDT
  16. MathJax Version: Duration of Game
    Updated Thursday, 28-Sep-2017 08:41:18 CDT
  17. PDF Version: The Ballot Theorem and the Reflection Principle
    Updated Tuesday, 31-Dec-2013 09:05:13 CST
  18. MathJax Version: The Ballot Theorem and the Reflection Principle
    Updated Friday, 11-Dec-2015 11:35:57 CST
  19. PDF Version: The Hitting Time Theorem
    Updated Tuesday, 05-Feb-2019 08:05:40 CST
  20. MathJax Version: The Hitting Time Theorem
    Updated Tuesday, 05-Feb-2019 08:05:39 CST
  21. PDF Version: Positive Walks
    Updated Wednesday, 24-Sep-2014 07:37:26 CDT
  22. MathJax Version: Positive Walks
    Updated Wednesday, 24-Sep-2014 07:39:53 CDT
  23. PDF Version: Arcsine Law
    Updated Tuesday, 10-Mar-2015 05:52:23 CDT
  24. MathJax Version: Arcsine Law
    Updated Tuesday, 10-Mar-2015 05:52:38 CDT
  25. PDF Version: Almost Sure Events
    Updated Monday, 25-Sep-2017 13:41:30 CDT
  26. MathJax Version: Almost Sure Events
    Updated Monday, 25-Sep-2017 13:41:34 CDT
  27. PDF Version: Strong Law of Large Numbers
    Updated Friday, 20-Oct-2017 05:53:50 CDT
  28. MathJax Version: Strong Law of Large Numbers
    Updated Friday, 20-Oct-2017 05:53:55 CDT
  29. PDF Version: Borel-Cantelli Lemmas with Examples
    Updated Tuesday, 10-Oct-2017 08:23:32 CDT
  30. MathJax Version: Borel-Cantelli Lemmas with Examples
    Updated Tuesday, 10-Oct-2017 08:23:38 CDT
  31. PDF Version: Law of the Iterated Logarithm
    Updated Thursday, 22-Mar-2018 14:06:20 CDT
  32. MathJax Version: Law of the Iterated Logarithm
    Updated Thursday, 22-Mar-2018 14:06:09 CDT
  33. PDF Version: Recurrence
    Updated Wednesday, 21-Feb-2018 05:45:44 CST
  34. MathJax Version: Recurrence
    Updated Wednesday, 21-Feb-2018 05:45:41 CST
  35. PDF Version: Recurrence in Higher Dimensions
    Updated Tuesday, 20-Mar-2018 12:08:59 CDT
  36. MathJax Version: Recurrence in Higher Dimensions
    Updated Tuesday, 20-Mar-2018 12:08:57 CDT
  37. PDF Version: Analytic Model of Coin Flipping
    Updated Friday, 15-Jun-2018 07:49:04 CDT
  38. MathJax Version: Analytic Model of Coin Flipping
    Updated Monday, 18-Jun-2018 07:34:03 CDT
  39. PDF Version: Analytic Proof of Central Limit Theorem
    Updated Friday, 12-Oct-2018 05:47:11 CDT
  40. MathJax Version: Analytic Proof of Central Limit Theorem
    Updated Friday, 12-Oct-2018 05:47:20 CDT
  41. PDF Version: Random Signs in Series, Random Series
    Updated Tuesday, 26-Jun-2018 07:49:56 CDT
  42. MathJax Version: Random Signs in Series, Random Series
    Updated Tuesday, 26-Jun-2018 07:58:50 CDT

Normal Random Variables, Gaussian Distributions

  1. The Sum of Independent Normal Random Variables is Normal
  2. Updated Saturday, 24-Jun-2017 09:15:41 CDT

Central Limit Theorem

  • PDF Version: The Sum of Independent Normal Random Variables is Normal
  • Updated Saturday, 24-Jun-2017 09:15:41 CDT

    Brownian Motion

    1. PDF Version: Introduction to Bernstein Polynomials and Brownian Motion
    2. PDF Version: Bernstein Polynomials and the Weierstrass Approximation Theorem
    3. PDF Version: Bernstein Motions and the convergence to Wiener measure

    Stochastic Calculus

    1. PDF Version: Examples of Stochastic Differential Equations

    Central Limit Theorem

    Large Deviations

    Markov Chains

    PDF Version: Review of Markov Processes
    Updated Thursday, 18-Apr-2019 11:04:21 CDT

    Classes of states and Stationary Distributions
    PDF Version: Classes of States and Stationary Distributions
    Updated Thursday, 18-Apr-2019 11:10:11 CDT

    PDF Version: Waiting Time to Absorption
    Updated Monday, 30-Jan-2017 05:59:17 CST

    Eigenvalues, Eigenvectors, and Normal Forms of Matrices
    PDF Version: Eigenvalues, Eigenvectors, and Normal Forms of Matrices
    Updated Friday, 28-May-2010 05:47:02 CDT


    PDF Version:

    Fastest Mixing Markov Chain
    Updated Friday, 16-Apr-2010 06:03:37 CDT
    PDF Version: Fastest Mixing Markov Chain
    Updated Friday, 16-Apr-2010 06:03:31 CDT

    PDF Version: Card Shuffling
    Updated Monday, 03-Feb-2020 07:34:01 CST


    Hidden Markov Models

    Examples of Hidden Markov Models
    Updated Wednesday, 19-Apr-2017 07:45:22 CDT
    PDF Version: Examples of Hidden Markov Models
    Updated Friday, 14-Apr-2017 08:18:38 CDT

    PDF : Talk/Slides about Examples of Hidden Markov Models
    Updated Friday, 14-Apr-2017 08:35:33 CDT

    Notation and Problems of Hidden Markov Models
    Updated Wednesday, 19-Apr-2017 07:46:02 CDT
    PDF Version: Examples of Hidden Markov Models
    Updated Friday, 14-Apr-2017 08:19:05 CDT

    PDF : Talk/Slides about Notation and Problems of Hidden Markov Models
    Updated Friday, 14-Apr-2017 08:35:36 CDT

    Algorithms for Hidden Markov Models
    Updated Tuesday, 09-May-2017 09:15:15 CDT
    PDF Version: Algorithms for Hidden Markov Models
    Updated Tuesday, 09-May-2017 09:15:11 CDT

    PDF : Talk/Slides about Algorithms for Hidden Markov Models
    Updated Friday, 03-Mar-2017 09:24:45 CST

    Applications, especially with Markov Chains

    Distinguishing a Biased Coin from a Fair Coin
    Updated Friday, 07-Sep-2018 08:00:19 CDT
    PDF Version: Distinguishing a Biased Coin from a Fair Coin
    Updated Friday, 07-Sep-2018 08:00:13 CDT

    Worst Case and Average Case Behavior of the Simplex Algorithm
    Updated Thursday, 20-Jan-2011 21:01:31 CST
    PDF Version: Worst Case and Average Case Behavior of the Simplex Algorithm
    Updated Thursday, 20-Jan-2011 21:01:27 CST

    Smoothed Analysis for Linear Optimization Algorithm
    Updated Thursday, 27-Jan-2011 20:52:43 CST
    PDF Version: Smoothed Analysis for Linear Optimization Algorithm
    Updated Thursday, 27-Jan-2011 20:52:41 CST

    Are the Digits in Mersenne Primes Random? A Probability Model.
    Updated Wednesday, 12-Oct-2016 08:36:11 CDT

    Martingales

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    Weak Law of Large Numbers

    Counting and Poisson Processes

    Conditional Probability and Expectation

    Online Tables for Probability and Cumulative Distributions

    Here are some links to Online Distribution Tables:
    1. Statsoft This site has several different distributions in a nice back-of-the-book-looking HTML format.
    2. David Lane, Rice University, Hyperstat This links to a Java applet that will compute the percentages for the normal distribution when given the Z values

    Annotated Bibliography

    A personal, eclectic and biased annotated bibliography of books and articles about probability, stochastic processes and mathematical finance.

    Outside Readings/Links:

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    [ Steve Dunbar's home page ] Steve Dunbar's Home Page, http://www.math.unl.edu/~sdunbar
    [ Email to Steve Dunbar ] Email to Steve Dunbar, sdunbar@unl.edu