Applied Probability

1. Discrete Probability.- 1.1. Applied Probability.- 1.2. Sample Spaces.- 1.3. Probability Distributions and Parameters.- 1.4. The Connection between Distributions and Sample Points: Random Variables.- 1.5. Events and Indicators.- 1.6. Mean and Variance.- 1.7. Calculation of the Mean and Variance.- 1.8. The Distribution Function.- 1.9. The Gamma Function and the Beta Function.- 1.10. The Negative Binomial Distribution.- 1.11. The Probability Generating Function.- 1.12. The Catalan Distribution.- 1.13. More about the p.g.f.; The Equation s = ?(s).- 1.14. Problems.- 2. Conditional Probability.- 2.1. Introduction. An Example.- 2.2. Conditional Probability and Bayes' Theorem.- 2.3. Conditioning.- 2.4. Independence and Bernoulli Trials.- 2.5. Moments, Distribution Functions, and Generating Functions.- 2.6. Convolutions and Sums of Random Variables.- 2.7. Computing Convolutions: Examples.- 2.8. Diagonal Distributions.- 2.9. Problems.- 3. Markov Chains.- 3.1. Introduction: Random Walk.- 3.2. Definitions.- 3.3. Matrix and Vector.- 3.4. The Transition Matrix and Initial Vector.- 3.5. The Higher-Order Transition Matrix: Regularity.- 3.6. Reducible Chains.- 3.7. Periodic Chains.- 3.8. Classification of States. Ergodic Chains.- 3.9. Finding Equilibrium Distributions-The Random Walk Revisited.- 3.10. A Queueing Model.- 3.11. The Ehrenfest Chain.- 3.12. Branching Chains.- 3.13. Probability of Extinction.- 3.14. The Gambler's Ruin.- 3.15. Probability of Ruin as Probability of Extinction.- 3.16. First-Passage Times.- 3.17. Problems.- 4. Continuous Probability Distributions.- 4.1. Examples.- 4.2. Probability Density Functions.- 4.3. Change of Variables.- 4.4. Convolutions of Density Functions.- 4.5. The Incomplete Gamma Function.- 4.6. The Beta Distribution and the Incomplete Beta Function.- 4.7. Parameter Mixing.- 4.8. Distribution Functions.- 4.9. Stieltjes Integration.- 4.10. The Laplace Transform.- 4.11. Properties of the Laplace Transform.- 4.12. Laplace Inversion.- 4.13. Random Sums.- 4.14. Problems.- 5. Continuous Time Processes.- 5.1. Introduction and Notation.- 5.2. Renewal Processes.- 5.3. The Poisson Process.- 5.4. Two-State Processes.- 5.5. Markov Processes.- 5.6. Equilibrium.- 5.7. The Method of Marks.- 5.8. The Markov Infinitesimal Matrix.- 5.9. The Renewal Function.- 5.10. The Gap Surrounding an Arbitrary Point.- 5.11. Counting Distributions.- 5.12. The Erlang Process.- 5.13. Displaced Gaps.- 5.14. Divergent Birth Processes.- 5.15. Problems.- 6. The Theory of Queues.- 6.1. Introduction and Classification.- 6.2. The M? / M? / 1 Queue: General Solution.- 6.3. The M? / M? / 1 Queue: Oversaturation.- 6.4. The M? / M? / 1 Queue: Equilibrium.- 6.5. The M? / M? / n Queue in Equilibrium: Loss Formula.- 6.6. The M? / G? / 1 Queue and the Imbedded Markov Chain.- 6.7. The Pollaczek-Khintchine Formula.- 6.8. Waiting Time.- 6.9. Virtual Queueing Time.- 6.10. The Equation y = xe?x.- 6.11. Busy Period: Borel's Method.- 6.12. The Busy Period Treated as a Branching Process: The M / G /1 Queue.- 6.13. The Continuous Busy Period and the M / G /1 Queue.- 6.14. Generalized Busy Periods.- 6.15. The G / M /1 Queue.- 616 Balking.- 6.17. Priority Service.- 6.18. Reverse-Order Service (LIFO).- 6.19. Problems.