Probability and Stochastic Processes

This textbook grew out of two classes that the author taught over the last decade at the University of Oklahoma's Tulsa campus. The students in these classes were doctoral candidates in engineering, and most specialized in some aspect of telecommunications. The doctoral program in engineering at the time required all students to obtain a minimum of six credit hours in graduate level work in either mathematics or physics. Many students chose one or both of these courses to fulfil this requirement. The University generously provided the author with a sabbatical to organize these notes into a textbook. The result is provided here on this website. Instructors at not-for-profit institutions may adopt this book and their students may access it free of charge, subject to the copyright notice below.

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Text and TeX layout and macros copyright 2017, William Ray
Please send comments to Bill@WilliamRay.net
Rights for non-commercial use are freely granted provided that this copyright notice is included.

Commercial users please contact the author at the above email address for licensing information. This textbook is provided "as-is" with no warranties, expressed or implied.
This book, TeX layout, and related macros, are subject to the GNU general public license as described in www.gnu.org/copyleft/gpl.html

Table of Contents

Chapter 1. Introduction, page 1
Chapter 2. Basic Probability, page 10
Chapter 3. The Multiplication Rule., page 20
Chapter 4. Combinations., page 28
Chapter 5. Probability Spaces, page 38
Chapter 6. Independence and Conditional Probability, page 45
Chapter 7. Random Variables, page 56
Chapter 8. Density Functions, page 62
Chapter 9. Examples of Discrete Random Variables, page 69
Chapter 10. Examples of Continuous Random Variables, page 83
Chapter 11. Jointly Distributed Discrete Random Variables, page 91
Chapter 12. Expectations: Discrete Random Variables, page 96
Chapter 13. Probability Generating Functions, page 106
Chapter 14. Jointly Distributed Continuous Random Variables, page 113
Chapter 15. Conditional Densities, page 128
Chapter 16. Expectations: Continuous Random Variables, page 136
Chapter 17. Moment Generating Functions, page 150
Chapter 18. Estimation, page 160
Chapter 19. Sampling Distributions., page 165
Chapter 20. Characteristic Functions, page 173
Chapter 21. Central Limit Theorem, page 179
Chapter 22. Applications to Polling, page 189
Chapter 23. Laws of Large Numbers, page 196
Chapter 24. Application to Learning Theory, page 210
Chapter 25. Two State Markov Chains, page 220
Chapter 26. Markov Chains -- Definitions and Examples, page 230
Chapter 27. Calculations with Transition Functions, page 242
Chapter 28. Transient and Recurrent States, page 252
Chapter 29. Absorption Probabilities, page 266
Chapter 30. Extinction Probabilities, page 281
Chapter 31. Stationary Distributions: Definitions and Examples, page 291
Chapter 32. Stationary Distributions: Results, page 306
Chapter 33. Convergence to the Stationary Distribution, page 325
Chapter 34. Poisson Processes, page 339
Chapter 35. Markov Pure Jump Processes, page 350
Chapter 36. Birth and Death Processes, page 368
Chapter 37. Queueing Processes, page 383
Chapter 38. Second Order Processes, page 386
Chapter 39. Integration and Differentiation of Processes, page 398
Chapter 40. Integration and Expectation of Processes, page 409
Chapter 41. Stochastic Differential Equations, page 417
Appendix A. Distributions., page 427
Chapter Index, page 432

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