Introduction to probability models / Sheldon M. Ross.

Format:

• Book

Author/Creator:

• Ross, Sheldon M.

Contributor:

• Class of 1939 Fund.

Language:

English

Subjects (All):

• Probabilities.

Physical Description:

xv, 693 pages : illustrations ; 24 cm

Edition:

Seventh edition.

Other Title:

Probability models

Place of Publication:

San Diego : Harcourt/Academic Press, [2000]

Summary:

The seventh edition of the successful "Introduction to Probability Models" introduces elementary probability theory and the stochastic processes and is particularly well-suited to those applying probability theory to the study of phenomena in engineering, management science, the physical and social sciences, and operations research. Skillfully organized, "Introduction to Probability Models" covers all essential topics. Sheldon Ross, a talented and prolific textbook author, distinguishes this carefully and substantially revised book by his effort to develop in students an intuitive, and therefore lasting, grasp of probability theory. The seventh edition includes many new examples and exercises, with the majority of the new exercises being less demanding of the student. In addition, the text introduces stochastic processes, stressing applications, in an easily understood manner. There is a comprehensive introduction to the applied models of probability that stresses intuition. Both students and professors will agree that this is the most solid and widely used text for probability theory. * Provides a detailed coverage of the Markov Chain Monte Carlo methods and Markov Chain covertimes* Gives a thorough presentation of k-record values and the surprising Ignatov's theorem* Includes examples relating to: "Random walks to circles," "The matching rounds problem," "The best prize problem" and many more* Contains a comprehensive appendix with the answers to approximately 100 exercises from throughout the text* Accompanied by a complete instructor's solutions manual with step-by-step solutions to all exercisesNEW TO THIS EDITION* Includes many new and easier examplesand exercises* Offers new material on utilizing probabilistic method in combinatorial optimization problems* Includes new material on suspended animation reliability models* Contains new material on random algorithms and cycles of random permutations

Contents:

• 1.2. Sample Space and Events 1
• 1.3. Probabilities Defined on Events 4
• 1.4. Conditional Probabilities 6
• 1.5. Independent Events 10
• 1.6. Bayes' Formula 12
• 2. Random Variables 23
• 2.1. Random Variables 23
• 2.2. Discrete Random Variables 27
• 2.2.1. The Bernoulli Random Variable 27
• 2.2.2. The Binomial Random Variable 28
• 2.2.3. The Geometric Random Variable 31
• 2.2.4. The Poisson Random Variable 31
• 2.3. Continuous Random Variables 33
• 2.3.1. The Uniform Random Variable 34
• 2.3.2. Exponential Random Variables 35
• 2.3.3. Gamma Random Variables 35
• 2.3.4. Normal Random Variables 36
• 2.4. Expectation of a Random Variable 37
• 2.4.1. The Discrete Case 37
• 2.4.2. The Continuous Case 40
• 2.4.3. Expectation of a Function of a Random Variable 42
• 2.5. Jointly Distributed Random Variables 46
• 2.5.1. Joint Distribution Functions 46
• 2.5.2. Independent Random Variables 50
• 2.5.3. Covariance and Variance of Sums of Random Variables 51
• 2.5.4. Joint Probability Distribution of Functions of Random Variables 59
• 2.6. Moment Generating Functions 62
• 2.6.1. The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population 70
• 2.7. Limit Theorems 73
• 2.8. Stochastic Processes 79
• 3. Conditional Probability and Conditional Expectation 93
• 3.2. The Discrete Case 93
• 3.3. The Continuous Case 98
• 3.4. Computing Expectations by Conditioning 101
• 3.5. Computing Probabilities by Conditioning 114
• 3.6. Some Applications 128
• 3.6.1. A List Model 128
• 3.6.2. A Random Graph 129
• 3.6.3. Uniform Priors, Polya's Urn Model, and Bose-Einstein Statistics 137
• 3.6.4. The k-Record Values of Discrete Random Variables 141
• 4. Markov Chains 163
• 4.2. Chapman-Kolmogorov Equations 166
• 4.3. Classification of States 168
• 4.4. Limiting Probabilities 178
• 4.5. Some Applications 188
• 4.5.1. The Gambler's Ruin Problem 188
• 4.5.2. A Model for Algorithmic Efficiency 192
• 4.5.3. Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem 194
• 4.6. Mean Time Spent in Transient States 200
• 4.7. Branching Processes 202
• 4.8. Time Reversible Markov Chains 205
• 4.9. Markov Chain Monte Carlo Methods 216
• 4.10. Markov Decision Processes 222
• 5. The Exponential Distribution and the Poisson Process 241
• 5.2. The Exponential Distribution 242
• 5.2.2. Properties of the Exponential Distribution 243
• 5.2.3. Further Properties of the Exponential Distribution 248
• 5.2.4. Convolutions of Exponential Random Variables 253
• 5.3. The Poisson Process 256
• 5.3.1. Counting Processes 256
• 5.3.2. Definition of the Poisson Process 258
• 5.3.3. Interarrival and Waiting Time Distributions 261
• 5.3.4. Further Properties of Poisson Processes 264
• 5.3.5. Conditional Distribution of the Arrival Times 270
• 5.3.6. Estimating Software Reliability 281
• 5.4. Generalizations of the Poisson Process 284
• 5.4.1. Nonhomogeneous Poisson Process 284
• 5.4.2. Compound Poisson Process 289
• 6. Continuous-Time Markov Chains 313
• 6.2. Continuous-Time Markov Chains 314
• 6.3. Birth and Death Processes 316
• 6.4. The Transition Probability Function P[subscript ij](t) 323
• 6.5. Limiting Probabilities 331
• 6.6. Time Reversibility 338
• 6.7. Uniformization 346
• 6.8. Computing the Transition Probabilities 349
• 7. Renewal Theory and Its Applications 363
• 7.2. Distribution of N(t) 365
• 7.3. Limit Theorems and Their Applications 368
• 7.4. Renewal Reward Processes 377
• 7.5. Regenerative Processes 386
• 7.5.1. Alternating Renewal Processes 389
• 7.6. Semi-Markov Processes 395
• 7.7. The Inspection Paradox 398
• 7.8. Computing the Renewal Function 400
• 7.9. Applications to Patterns 403
• 7.9.1. Patterns of Discrete Random Variables 404
• 7.9.2. The Expected Time to a Maximal Run of Distinct Values 410
• 7.9.3. Increasing Runs of Continuous Random Variables 412
• 8. Queueing Theory 427
• 8.2. Preliminaries 428
• 8.2.1. Cost Equations 429
• 8.2.2. Steady-State Probabilities 430
• 8.3. Exponential Models 432
• 8.3.1. A Single-Server Exponential Queueing System 432
• 8.3.2. A Single-Server Exponential Queueing System Having Finite Capacity 438
• 8.3.3. A Shoeshine Shop 442
• 8.3.4. A Queueing System with Bulk Service 444
• 8.4. Network of Queues 447
• 8.4.1. Open Systems 447
• 8.4.2. Closed Systems 452
• 8.5. The System M/G/1 458
• 8.5.1. Preliminaries: Work and Another Cost Identity 458
• 8.5.2. Application of Work to M/G/1 459
• 8.5.3. Busy Periods 460
• 8.6. Variations on the M/G/1 461
• 8.6.1. The M/G/1 with Random-Sized Batch Arrivals 461
• 8.6.2. Priority Queues 463
• 8.6.3. An M/G/1 Optimization Example 466
• 8.7. The Model G/M/1 470
• 8.7.1. The G/M/1 Busy and Idle Periods 475
• 8.8. A Finite Source Model 475
• 8.9. Multiserver Queues 479
• 8.9.1. Erlang's Loss System 479
• 8.9.2. The M/M/k Queue 481
• 8.9.3. The G/M/k Queue 481
• 8.9.4. The M/G/k Queue 483
• 9. Reliability Theory 499
• 9.2. Structure Functions 500
• 9.2.1. Minimal Path and Minimal Cut Sets 502
• 9.3. Reliability of Systems of Independent Components 506
• 9.4. Bounds on the Reliability Function 510
• 9.4.1. Method of Inclusion and Exclusion 511
• 9.4.2. Second Method for Obtaining Bounds on r(p) 519
• 9.5. System Life as a Function of Component Lives 521
• 9.6. Expected System Lifetime 529
• 9.6.1. An Upper Bound on the Expected Life of a Parallel System 533
• 9.7. Systems with Repair 535
• 9.7.1. A Series Model with Suspended Animation 539
• 10. Brownian Motion and Stationary Processes 549
• 10.1. Brownian Motion 549
• 10.2. Hitting Times, Maximum Variable, and the Gambler's Ruin Problem 553
• 10.3. Variations on Brownian Motion 554
• 10.3.1. Brownian Motion with Drift 554
• 10.3.2. Geometric Brownian Motion 555
• 10.4. Pricing Stock Options 556
• 10.4.1. An Example in Options Pricing 556
• 10.4.2. The Arbitrage Theorem 558
• 10.4.3. The Black-Scholes Option Pricing Formula 561
• 10.5. White Noise 567
• 10.6. Gaussian Processes 569
• 10.7. Stationary and Weakly Stationary Processes 572
• 10.8. Harmonic Analysis of Weakly Stationary Processes 577
• 11. Simulation 585
• 11.2. General Techniques for Simulating Continuous Random Variables 590
• 11.2.1. The Inverse Transformation Method 590
• 11.2.2. The Rejection Method 591
• 11.2.3. The Hazard Rate Method 595
• 11.3. Special Techniques for Simulating Continuous Random Variables 598
• 11.3.1. The Normal Distribution 598
• 11.3.2. The Gamma Distribution 602
• 11.3.3. The Chi-Squared Distribution 602
• 11.3.4. The Beta (n, m) Distribution 603
• 11.3.5. The Exponential Distribution
• The Von Neumann Algorithm 604
• 11.4. Simulating from Discrete Distributions 606
• 11.4.1. The Alias Method 610
• 11.5. Stochastic Processes 613
• 11.5.1. Simulating a Nonhomogeneous Poisson Process 615
• 11.5.2. Simulating a Two-Dimensional Poisson Process 621
• 11.6. Variance Reduction Techniques 624
• 11.6.1. Use of Antithetic Variables 625
• 11.6.2. Variance Reduction by Conditioning 629
• 11.6.3. Control Variates 633
• 11.6.4. Importance Sampling 634
• 11.7. Determining the Number of Runs 639.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1939 Fund.

ISBN:

0125984758

OCLC:

43070170

Online:

The Class of 1939 Fund Home Page