Introduction to probability models:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
San Diego [u.a.]
Acad. Press
1997
|
| Ausgabe: | 6. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 669 S. |
| ISBN: | 0125984707 |
Internformat
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| 245 | 1 | 0 | |a Introduction to probability models |c Sheldon M. Ross |
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Datensatz im Suchindex
| _version_ | 1804125961423159296 |
|---|---|
| adam_text | Contents
Preface to the Sixth Edition xi
Preface to the Fifth Edition xiii
1. Introduction to Probability Theory
1.1. Introduction 1
1.2. Sample Space and Events 1
1.3. Probabilities Defined on Events 4
1.4. Conditional Probabilities 7
1.5. Independent Events 10
1.6. Bayes Formula 12
Exercises 15
References 20
2. Random Variables
2.1. Random Variables 21
2.2. Discrete Random Variables 25
2.2.1. The Bernoulli Random Variable 26
2.2.2. The Binomial Random Variable 26
2.2.3. The Geometric Random Variable 29
2.2.4. The Poisson Random Variable 30
2.3. Continuous Random Variables 31
2.3.1. The Uniform Random Variable 32
2.3.2. Exponential Random Variables 34
2.3.3. Gamma Random Variables 34
2.3.4. Normal Random Variables 34
2.4. Expectation of a Random Variable 36
2.4.1. The Discrete Case 36
2.4.2. The Continuous Case 39
2.4.3. Expectation of a Function of a Random Variable 40
v
vi Contents
2.5. Jointly Distributed Random Variables 44
2.5.1. Joint Distribution Functions 44
2.5.2. Independent Random Variables 48
2.5.3. Covariance and Variance of Sums of Random Variables 49
2.5.4. Joint Probability Distribution of Functions
of Random Variables 58
2.6. Moment Generating Functions 60
2.6.1. The Joint Distribution of the Sample Mean and
Sample Variance from a Normal Population 68
2.7. Limit Theorems 71
2.8. Stochastic Processes 77
Exercises 79
References 90
3. Conditional Probability and Conditional Expectation
3.1. Introduction 91
3.2. The Discrete Case 91
3.3. The Continuous Case 96
3.4. Computing Expectations by Conditioning 99
3.5. Computing Probabilities by Conditioning 111
3.6. Some Applications 124
3.6.1. A List Model 124
3.6.2. A Random Graph 126
3.6.3. Uniform Priors, Polya s Urn Model, and
Bose Einstein Statistics 133
3.6.4. The Ar Record Values of Discrete Random Variables 137
Exercises 141
4. Markov Chains
4.1. Introduction 157
4.2. Chapman Kolmogorov Equations 160
4.3. Classification of States 163
4.4. Limiting Probabilities 172
4.5. Some Applications 183
4.5.1. The Gambler s Ruin Problem 183
4.5.2. A Model for Algorithmic Efficiency 186
4.5.3. Using a Random Walk to Analyze a Probabilistic
Algorithm for the Satisfiability Problem 189
4.6. Mean Time Spent in Transient States 195
4.7. Branching Processes 197
4.8. Time Reversible Markov Chains 201
4.9. Markov Chain Monte Carlo Methods 211
Contents vii
4.10. Markov Decision Processes 217
Exercises 221
References 234
5. The Exponential Distribution and the Poisson Process
5.1. Introduction 235
5.2. The Exponential Distribution 236
5.2.1. Definition 236
5.2.2. Properties of the Exponential Distribution 237
5.2.3. Further Properties of the Exponential Distribution 242
5.2.4. Convolutions of Exponential Random Variables 245
5.3. The Poisson Process 249
5.3.1. Counting Processes 249
5.3.2. Definition of the Poisson Process 250
5.3.3. Interarrival and Waiting Time Distributions 255
5.3.4. Further Properties of Poisson Processes 257
5.3.5. Conditional Distribution of the Arrival Times 263
5.3.6. Estimating Software Reliability 275
5.4. Generalizations of the Poisson Process 277
5.4.1. Nonhomogeneous Poisson Process 277
5.4.2. Compound Poisson Process 281
Exercises 287
References 301
6. Continuous Time Markov Chains
6.1. Introduction 303
6.2. Continuous Time Markov Chains 304
6.3. Birth and Death Processes 306
6.4. The Transition Probability Function Pv(t) 313
6.5. Limiting Probabilities 322
6.6. Time Reversibility 329
6.7. Uniformization 335
6.8. Computing the Transition Probabilities 338
Exercises 341
References 350
7. Renewal Theory and Its Applications
7.1. Introduction 351
7.2. Distribution of N(t) 353
7.3. Limit Theorems and Their Applications 357
viii Contents
7.4. Renewal Reward Processes 366
7.5. Regenerative Processes 373
7.5.1. Alternating Renewal Processes 374
7.6. Semi Markov Processes 379
7.7. The Inspection Paradox 382
7.8. Computing the Renewal Function 384
7.9. Applications to Patterns 387
7.9.1. Patterns of Discrete Random Variables 388
7.9.2. The Expected Time to a Maximal Run of
Distinct Values 395
7.9.3. Increasing Runs of Continuous Random Variables 397
Exercises 398
References 409
8. Queueing Theory
8.1. Introduction 411
8.2. Preliminaries 412
8.2.1. Cost Equations 412
8.2.2. Steady State Probabilities 414
8.3. Exponential Models 416
8.3.1. A Single Server Exponential Queueing System 416
8.3.2. A Single Server Exponential Queueing System
Having Finite Capacity 423
8.3.3. A Shoeshine Shop 426
8.3.4. A Queueing System with Bulk Service 429
8.4. Network of Queues 432
8.4.1. Open Systems 432
8.4.2. Closed Systems 437
8.5. The System M/G/l 442
8.5.1. Preliminaries: Work and Another Cost Identity 442
8.5.2. Application of Work to M/G/l 443
8.5.3. Busy Periods 444
8.6. Variations on the M/G/l 446
8.6.1. The M/G/l with Random Sized Batch Arrivals 446
8.6.2. Priority Queues 448
8.7. The Model G/M/l 451
8.7.1. The G/M/l Busy and Idle Periods 455
8.8. Multiserver Queues 456
8.8.1. Erlang s Loss System 456
8.8.2. The M/M/k Queue 458
Contents ix
8.8.3. The G/M/k Queue 458
8.8.4. The M/G/k Queue 460
Exercises 462
References 473
9. Reliability Theory
9.1. Introduction 475
9.2. Structure Functions 476
9.2.1. Minimal Path and Minimal Cut Sets 478
9.3. Reliability of Systems of Independent Components 482
9.4. Bounds on the Reliability Function 486
9.4.1. Method of Inclusion and Exclusion 487
9.4.2. Second Method for Obtaining Bounds on rip) 495
9.5. System Life as a Function of Component Lives 497
9.6. Expected System Lifetime 505
9.6.1. An Upper Bound on the Expected Life of a
Parallel System 509
9.7. Systems with Repair 511
Exercises 515
References 522
10. Brownian Motion and Stationary Processes
10.1. Brownian Motion 523
10.2. Hitting Times, Maximum Variable, and the Gambler s
Ruin Problem 527
10.3. Variations on Brownian Motion 529
10.3.1. Brownian Motion with Drift 529
10.3.2. Geometric Brownian Motion 529
10.4. Pricing Stock Options 530
10.4.1. An Example in Options Pricing 530
10.4.2. The Arbitrage Theorem 533
10.4.3. The Black Scholes Option Pricing Formula 536
10.5. White Noise 541
10.6. Gaussian Processes 543
10.7. Stationary and Weakly Stationary Processes 546
10.8. Harmonic Analysis of Weakly Stationary Processes 551
Exercises 553
References 558
x Contents
11. Simulation
11.1. Introduction 559
11.2. General Techniques for Simulating Continuous
Random Variables 564
11.2.1. The Inverse Transformation Method 564
11.2.2. The Rejection Method 565
11.2.3. The Hazard Rate Method 569
11.3. Special Techniques for Simulating Continuous
Random Variables 572
11.3.1. The Normal Distribution 572
11.3.2. The Gamma Distribution 576
11.3.3. The Chi Squared Distribution 576
11.3.4. The Beta (n, m) Distribution 577
11.3.5. The Exponential Distribution—The Von Neumann
Algorithm 578
11.4. Simulating from Discrete Distributions 580
11.4.1. The Alias Method 584
11.5. Stochastic Processes 587
11.5.1. Simulating a Nonhomogeneous Poisson Process 589
11.5.2. Simulating a Two Dimensional Poisson Process 595
11.6. Variance Reduction Techniques 598
11.6.1. Use of Antithetic Variables 599
11.6.2. Variance Reduction by Conditioning 602
11.6.3. Control Variates 606
11.6.4. Importance Sampling 608
11.7. Determining the Number of Runs 613
Exercises 613
References 622
Appendix: Solutions to Starred Exercises 623
Index 663
|
| any_adam_object | 1 |
| author | Ross, Sheldon M. 1943- |
| author_GND | (DE-588)123762235 |
| author_facet | Ross, Sheldon M. 1943- |
| author_role | aut |
| author_sort | Ross, Sheldon M. 1943- |
| author_variant | s m r sm smr |
| building | Verbundindex |
| bvnumber | BV011443964 |
| classification_rvk | QH 170 SK 800 |
| classification_tum | MAT 600f |
| ctrlnum | (OCoLC)247680515 (DE-599)BVBBV011443964 |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 6. ed. |
| format | Book |
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| isbn | 0125984707 |
| language | English |
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| spelling | Ross, Sheldon M. 1943- Verfasser (DE-588)123762235 aut Introduction to probability models Sheldon M. Ross 6. ed. San Diego [u.a.] Acad. Press 1997 XIV, 669 S. txt rdacontent n rdamedia nc rdacarrier Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Modell (DE-588)4039798-1 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Modell (DE-588)4039798-1 s DE-604 Stochastischer Prozess (DE-588)4057630-9 s Stochastisches Modell (DE-588)4057633-4 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s 1\p DE-604 Mathematisches Modell (DE-588)4114528-8 s 2\p DE-604 3\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007697366&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ross, Sheldon M. 1943- Introduction to probability models Stochastischer Prozess (DE-588)4057630-9 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Stochastisches Modell (DE-588)4057633-4 gnd Modell (DE-588)4039798-1 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
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| title | Introduction to probability models |
| title_auth | Introduction to probability models |
| title_exact_search | Introduction to probability models |
| title_full | Introduction to probability models Sheldon M. Ross |
| title_fullStr | Introduction to probability models Sheldon M. Ross |
| title_full_unstemmed | Introduction to probability models Sheldon M. Ross |
| title_short | Introduction to probability models |
| title_sort | introduction to probability models |
| topic | Stochastischer Prozess (DE-588)4057630-9 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Stochastisches Modell (DE-588)4057633-4 gnd Modell (DE-588)4039798-1 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Stochastischer Prozess Wahrscheinlichkeitstheorie Stochastisches Modell Modell Wahrscheinlichkeitsrechnung Mathematisches Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007697366&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT rosssheldonm introductiontoprobabilitymodels |