Introduction to Probability Models, Ninth Edition

Ross's classic bestseller, Introduction to Probability Models, has been used extensively by professionals and as the primary text for a first undergraduate course in applied probability. It provides an introduction to elementary probability theory and stochastic processes, and shows how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management science, the physical and social sciences, and operations research. With the addition of several new sections relating to actuaries, this text is highly recommended by the Society of Actuaries. A new section (3.7) on COMPOUND RANDOM VARIABLES, that can be used to establish a recursive formula for computing probability mass functions for a variety of common compounding distributions. A new section (4.11) on HIDDDEN MARKOV CHAINS, including the forward and backward approaches for computing the joint probability mass function of the signals, as well as the Viterbi algorithm for determining the most likely sequence of states. Simplified Approach for Analyzing Nonhomogeneous Poisson processes Additional results on queues relating to the (a) conditional distribution of the number found by an M/M/1 arrival who spends a time t in the system,; (b) inspection paradox for M/M/1 queues (c) M/G/1 queue with server breakdown Many new examples and exercises. Contents......Page 3 1.2. Sample Space and Events......Page 16 1.3. Probabilities Defined on Events......Page 19 1.4. Conditional Probabilities......Page 22 1.5. Independent Events......Page 25 1.6. Bayes' Formula......Page 27 Exercises......Page 30 References......Page 36 2.1. Random Variables......Page 37 2.2. Discrete Random Variables......Page 41 2.3. Continuous Random Variables......Page 48 2.4. Expectation of a Random Variable......Page 52 2.5. Jointly Distributed Random Variables......Page 61 2.6. Moment Generating Functions......Page 78 2.7. Limit Theorems......Page 91 2.8. Stochastic Processes......Page 97 Exercises......Page 99 References......Page 110 3.2. The Discrete Case......Page 111 3.3. The Continuous Case......Page 116 3.4. Computing Expectations by Conditioning......Page 119 3.5. Computing Probabilities by Conditioning......Page 134 3.6. Some Applications......Page 151 3.7. An Identity for Compound Random Variables......Page 172 Exercises......Page 179 4.1. Introduction......Page 199 4.2. Chapman-Kolmogorov Equations......Page 203 4.3. Classification of States......Page 207 4.4. Limiting Probabilities......Page 218 4.5. Some Applications......Page 231 4.6. Mean Time Spent in Transient States......Page 244 4.7. Branching Processes......Page 247 4.8. Time Reversible Markov Chains......Page 250 4.9. Markov Chain Monte Carlo Methods......Page 261 4.10. Markov Decision Processes......Page 266 4.11. Hidden Markov Chains......Page 270 Exercises......Page 277 References......Page 294 5.1. Introduction......Page 295 5.2. The Exponential Distribution......Page 296 5.3. The Poisson Process......Page 316 5.4. Generalizations of the Poisson Process......Page 344 Exercises......Page 360 References......Page 378 6.1. Introduction......Page 379 6.2. Continuous-Time Markov Chains......Page 380 6.3. Birth and Death Processes......Page 382 6.4. The Transition Probability Function Pij(t)......Page 389 6.5. Limiting Probabilities......Page 398 6.6. Time Reversibility......Page 406 6.7. Uniformization......Page 415 6.8. Computing the Transition Probabilities......Page 418 Exercises......Page 421 References......Page 429 7.1. Introduction......Page 430 7.2. Distribution of N(t)......Page 432 7.3. Limit Theorems and Their Applications......Page 436 7.4. Renewal Reward Processes......Page 446 7.5. Regenerative Processes......Page 455 7.6. Semi-Markov Processes......Page 465 7.7. The Inspection Paradox......Page 468 7.8. Computing the Renewal Function......Page 471 7.9. Applications to Patterns......Page 474 7.10. The Insurance Ruin Problem......Page 486 Exercises......Page 492 References......Page 505 8.1. Introduction......Page 506 8.2. Preliminaries......Page 507 8.3. Exponential Models......Page 512 8.4. Network of Queues......Page 530 8.5. The System M/G/1......Page 541 8.6. Variations on the M/G/1......Page 544 8.7. The Model G/M/1......Page 556 8.8. A Finite Source Model......Page 562 8.9. Multiserver Queues......Page 565 Exercises......Page 571 References......Page 583 9.2. Structure Functions......Page 584 9.3. Reliability of Systems of Independent Components......Page 591 9.4. Bounds on the Reliability Function......Page 596 9.5. System Life as a Function of Component Lives......Page 608 9.6. Expected System Lifetime......Page 617 9.7. Systems with Repair......Page 623 Exercises......Page 630 References......Page 637 10.1. Brownian Motion......Page 638 10.2. Hitting Times, Maximum Variable, and the Gambler's Ruin Problem......Page 642 10.3. Variations on Brownian Motion......Page 644 10.4. Pricing Stock Options......Page 645 10.5. White Noise......Page 657 10.6. Gaussian Processes......Page 659 10.7. Stationary and Weakly Stationary Processes......Page 662 10.8 Harmonic Analysis of Weakly Stationary Processes......Page 667 Exercises......Page 670 References......Page 675 11.1. Introduction......Page 676 11.2. General Techniques for Simulating Continuous Random Variables......Page 681 11.3. Special Techniques for Simulating Continuous Random Variables......Page 690 11.4. Simulating from Discrete Distributions......Page 698 11.5. Stochastic Processes......Page 705 11.6. Variance Reduction Techniques......Page 716 11.8. Coupling from the Past......Page 733 Exercises......Page 736 References......Page 744 Appendix: Solutions to Starred Exercises......Page 745 Index......Page 786 Introduction to Probability Models, Ninth Edition, is the primary text for a first undergraduate course in applied probability. This updated edition of Ross's classic bestseller provides an introduction to elementary probability theory and stochastic processes, and shows how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management science, the physical and social sciences, and operations research. With the addition of several new sections relating to actuaries, this text is highly recommended by the Society of Actuaries. This book now contains a new section on compound random variables that can be used to establish a recursive formula for computing probability mass functions for a variety of common compounding distributions; a new section on hiddden Markov chains, including the forward and backward approaches for computing the joint probability mass function of the signals, as well as the Viterbi algorithm for determining the most likely sequence of states; and a simplified approach for analyzing nonhomogeneous Poisson processes. There are also additional results on queues relating to the conditional distribution of the number found by an M/M/1 arrival who spends a time t in the system; inspection paradox for M/M/1 queues; and M/G/1 queue with server breakdown. Furthermore, the book includes new examples and exercises, along with compulsory material for new Exam 3 of the Society of Actuaries. This book is essential reading for professionals and students in actuarial science, engineering, operations research, and other fields in applied probability. A new section (3.7) on COMPOUND RANDOM VARIABLES, that can be used to establish a recursive formula for computing probability mass functions for a variety of common compounding distributions. A new section (4.11) on HIDDDEN MARKOV CHAINS, including the forward and backward approaches for computing the joint probability mass function of the signals, as well as the Viterbi algorithm for determining the most likely sequence of states. Simplified Approach for Analyzing Nonhomogeneous Poisson processes Additional results on queues relating to the (a) conditional distribution of the number found by an M/M/1 arrival who spends a time t in the system,; (b) inspection paradox for M/M/1 queues (c) M/G/1 queue with server breakdown Many new examples and exercises. Ross's classic bestseller, Introduction to Probability Models, has been used extensively by professionals and as the primary text for a first undergraduate course in applied probability. It provides an introduction to elementary probability theory and stochastic processes, and shows how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management science, the physical and social sciences, and operations research. With the addition of several new sections relating to actuaries, this text is highly recommended by the Society of Actuaries.

A new section (3.7) on COMPOUND RANDOM VARIABLES, that can be used to establish a recursive formula for computing probability mass functions for a variety of common compounding distributions.

A new section (4.11) on HIDDDEN MARKOV CHAINS, including the forward and backward approaches for computing the joint probability mass function of the signals, as well as the Viterbi algorithm for determining the most likely sequence of states.

Simplified Approach for Analyzing Nonhomogeneous Poisson processes

Additional results on queues relating to the
(a) conditional distribution of the number found by an M/M/1 arrival who spends a time t in the system,;
(b) inspection paradox for M/M/1 queues
(c) M/G/1 queue with server breakdown


Many new examples and exercises. The Seventh Edition of Ross' Intorduction to Probability Models represents the continuing convergence of this best-selling book with the widening indispensability of probability in pure and applied science.Revised and updated, Introduction to Probability Models is particularly well suited to those seeking an understanding of how probability theory and stochastic processes apply to phenomena in such fields as engineering, management science, the physical and social sciences, and operations research.While retaining its focus on elementary probability and stochastic processes, this edition's significant revisions include:\* Nearly 600 new or updated exercises, with over 100 solutions provided\* New derivations for the Poisson and nonhomogeneous Poisson processes\* Optimization of a single server, general service time queue\* Analysis of a series structure reliability model in which components enter a state of suspended animation upon cohort failureSheldon M. Ross has published numerous textbooks and technical articles in the areas of statistics and applied probability. Professor Ross is the founding and continuing editor of the journal Probability in the Engineering and Informational Sciences, published by Cambridge University Press. He is a fellow of the Institute of Mathematical Statistics and a recipient of the Humboldt U.S. Senior Scientist Award. "Ross's classic bestseller, Introduction to Probability Models, has been used extensively by professors as the primary text for a first undergraduate course in applied probability. It provides an Introduction to elementary probability theory and stochastic processes, and shows how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management science, the physical and social sciences, and operations research. With the addition of several new sections relating to actuaries, this text is highly recommended by the Society of Actuaries. The tenth edition contains several sections covered in the new exams."--Jacket.