STAT/MATH 6402, Advanced Probability II, Winter 2005

Instructor: Prof. Jaimie Kwon (Homepage)

Lecture: MW ScN 207, 4-5:50 PM

The book website

Syllabus

Table of Contents of the textbook (for Project Assignments)

Contents
Preface
1. Introduction to Probability Theory
1.1. Introduction
1.2. Sample Space and Events
1.3. Probabilities Defined on Events
1.4. Conditional Probabilities
1.5. Independent Events
1.6. Bayes' Formula
Exercises
References
2. Random Variables
2.1. Random Variables
2.2. Discrete Random Variables
2.3. Continuous Random Variables
2.4. Expectation of a Random Variable
2.5. Jointly Distributed Random Variables
2.6. Moment Generating Functions
2.7. Limit Theorems
2.8. Stochastic Processes
3. Conditional Probability and Conditional
Expectation
3.1. Introduction
3.2. The Discrete Case
3.3. The Continuous Case
3.4. Computing Expectations by Conditioning
3.5. Computing Probabilities by Conditioning
3.6. Some Applications
Exercises
4. Markov Chains
4.1. Introduction
4.2. Chapman-Kolmogorov Equations
4.3. Classification of States
4.4. Limiting Probabilities
4.5. Some Applications
4.6. Mean Time Spent in Transient States
4.7. Branching Processes
4.8. Time Reversible Markov Chains
4.9. Markov Chain Monte Carlo Methods
4.10. Markov Decision Processes
Exercises
References
5. The Exponential Distribution and the Poisson
Process
5.1. Introduction
5.2. The Exponential Distribution
5.3. The Poisson Process
5.4. Generalizations of the Poisson Process
Exercises
References
6. Continuous-Time Markov Chains
6.1. Introduction
6.2. Continuous-Time Markov Chains
6.3. Birth and Death Processes
6.4. The Transition Probability Function Pij (t)
6.5. Limiting Probabilities
6.6. Time Reversibility
6.7. Uniformization
6.8. Computing the Transition Probabilities
Exercises
References
7. Renewal Theory and Its Applications
7.1. Introduction
7.2. Distribution of N(t)
7.3. Limit Theorems and Their Applications
7.4. Renewal Reward Processes
7.5. Regenerative Processes
7.6. Semi-Markov Processes
7.7. The Inspection Paradox
7.8. Computing the Renewal Function
7.9. Applications to Patterns
7.10. The Insurance Ruin Problem
Exercises
References
8. Queueing Theory
8.1. Introduction
8.2. Preliminaries
8.3. Exponential Models
8.4. Network of Queues
8.5. The System M/G/1
8.6. Variations on the M/G/1
8.7. The Model G/M/1
8.8. A Finite Source Model
8.9. Multiserver Queues
Exercises
References
9. Reliability Theory
9.1. Introduction
9.2. Structure Functions
9.3. Reliability of Systems of Independent Components
9.4. Bounds on the Reliability Function
9.5. System Life as a Function of Component Lives
9.6. Expected System Lifetime
9.7. Systems with Repair
Exercises
References
10. Brownian Motion and Stationary Processes
10.1. Brownian Motion
10.2. Hitting Times, Maximum Variable, and the Gambler's Ruin Problem
10.3. Variations on Brownian Motion
10.4. Pricing Stock Options
10.5. White Noise
10.6. Gaussian Processes
10.7. Stationary andWeakly Stationary Processes
10.8. Harmonic Analysis of Weakly Stationary Processes
Exercises
References
11. Simulation
11.1. Introduction
11.2. General Techniques for Simulating Continuous Random Variables
11.3. Special Techniques for Simulating Continuous Random Variables
11.4. Simulating from Discrete Distributions
11.5. Stochastic Processes
11.6. Variance Reduction Techniques
11.7. Determining the Number of Runs
11.8. Coupling from the Past
Exercises
References
Appendix: Solutions to Starred Exercises
Index

Last updated 03/07/2006