Intermediate Probability is not a formal ETSU class (not yet, at least). This would be a cross-listed 4000/5000 level class with potential catalog description: "Provides an intermediate course in probability theory beyond the introductory level. No knowledge of measure theory is assumed. Topics include multivariate random variables, conditional distributions and expectation, regression, transforms, order statistics, multivariate normal distributions, convergence concepts, and Poisson processes." Prerequisites are Linear Algebra (MATH 2010) and Foundations of Probability and Statistics-Calculus Based (MATH 2050). Good companion classes to Intermediate Probability Theory are Mathematical Statistics 1 (STAT 4047/5057) and Mathematical Statistics 2 (STAT 4057/5057). The main body of this course is Chapters 1 through 7 of the text.

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes have not been classroom tested and may have typographical errors.

• Introduction.
• Chapter 1. Multivariate Random Variables.
• Chapter 2. Conditioning.
• Chapter 3. Transforms.
• Chapter 4. Order Statistics.
• Chapter 5. The Multivariate Normal Distribution.
• Chapter 6. Convergence.
• Chapter 7. An Outlook on Further Topics.
• Chapter 8. The Poisson Process.
• Appendices.

Introduction

• 0.1. Models. Section 0.1 notes
• 0.2. The Probability Space. Section 0.2 notes (This section includes biographical information on Andrey Kolmogorov.)
• 0.3. Independence and Conditional Probabilities. Section 0.3 notes (This section includes brief biographical information on Thomas Bayes.)
• 0.4. Random Variables. Section 0.4 notes
• 0.5. Expectation, Variance, and Moments. Section 0.5 notes
• 0.6. Joint Distributions and Independence. Section 0.6 notes
• 0.7. Sums of Random Variables, Covariance, Correlation. Section 0.7 notes
• 0.8. Limit Theorems. Section 0.8 notes
• 0.9. Stochastic Processes. Section 0.9 notes
• 0.10. The Contents of the Book.
• Study Guide 0

1. Multivariate Random Variables.

• 1.1. Introduction. Section 1.1 notes
• 1.2. Functions of Random Variables.
• Study Guide 1.

2. Conditioning.

• 2.1. Conditional Distributions.
• 2.2. Conditional Expectation.
• 2.3. Distributions with Random Parameters.
• 2.4. The Bayesian Approach.
• 2.5. Regression and Prediction.
• Study Guide 2.

3. Transforms.

• 3.1. Introduction.
• 3.2. The Probability Generating Function.
• 3.3. The Moment Generating Function.
• 3.4. The Characteristic Function.
• 3.5. Distributions with Random Parameters.
• 3.6. Sums of a Random Number of Random Variables.
• 3.7. Branching Processes.
• Study Guide 3.

4. Ordering Statistics.

• 4.1. One-Dimensional Results.
• 4.2. The Joint Distribution of the Extremes.
• 4.3. The Joint Distribution of the Order Statistic.
• Study Guide 4.

5. The Multivariate Normal Distribution.

• 5.1. Preliminaries from Linear Algebra.
• 5.2. The Covariance Matrix.
• 5.3. A First Definition.
• 5.4. The Characeristic Function: Another Definition.
• 5.5. The Density: A Third Definition.
• 5.6. Conditional Distributions.
• 5.7. Independence.
• 5.8. Linear Transformations.
• 5.9. Quadratic Forms and Cochran's Theorem.
• Study Guide 5.

6. Convergence.

• 6.1. Definitions.
• 6.2. Uniqueness.
• 6.3. Relations Between the Convergence Concepts.
• 6.4. Convergence via Transforms.
• 6.5. The Law of Large Numbers and the Central Limit Theorem.
• 6.6. Convergence of Sums of Sequences of Random Variables.
• 6.7. The Galton-Watson Process Revisited.
• Study Guide 6.

7. An Outlook on Further Topics.

• 7.1. Extensions of the Main Limit Theorems.
• 7.2. Stable Distributions.
• 7.3. Domains of Attraction.
• 7.4. Uniform Integrability.
• 7.5. An Introduction to Extreme Value Theory.
• 7.6. Records.
• 7.7. The Borel-Cantelli Lemmmas.
• 7.8. Martingales.
• Study Guide 7.

8. The Poisson Process.

• 8.1. Introduction and Definitions.
• 8.2. Restarted Poisson Processes.
• 8.3. Conditioning on the Number of Occurences in an Interval.
• 8.4. Conditioning on Occurence Times.
• 8.5. Several Independent Poisson Processes.
• 8.6. Thinning of Poisson Processes.
• 8.7. The The Compound Poisson Process.
• 8.8. Some Further Generalizations and Remarks.
• Study Guide 8.

Appendices.

• A. Suggestions for Further Reading.
• B. Some Distributions and Their Characteristics.
• C. Answers to Problems.

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