An introduction to stochastic modeling:
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| Previous Title: | Taylor, Howard M. An Introduction to stochastic modeling |
|---|---|
| Main Authors: | , |
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier [u.a.]
2011
|
| Edition: | 4. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 563 S. graph. Darst. 23 cm |
| ISBN: | 9780123814166 |
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| _version_ | 1804143694785282048 |
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| adam_text | Titel: An introduction to stochastic modeling
Autor: Pinsky, Mark A.
Jahr: 2011
Contents
XI
xiii
xv
xvii
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the First Edition
To the Instructor
Acknowledgments xix
1 Introduction 1
1.1 Stochastic Modeling 1
1.1.1 Stochastic Processes 4
1.2 Probability Review 4
1.2.1 Events and Probabilities 4
1.2.2 Random Variables 5
1.2.3 Moments and Expected Values 7
1.2.4 Joint Distribution Functions 8
1.2.5 Sums and Convolutions 10
1.2.6 Change of Variable 10
1.2.7 Conditional Probability 11
1.2.8 Review of Axiomatic Probability Theory 12
1.3 The Major Discrete Distributions 19
1.3.1 Bernoulli Distribution 20
1.3.2 Binomial Distribution 20
1.3.3 Geometric and Negative Binominal Distributions 21
1.3.4 The Poisson Distribution 22
1.3.5 The Multinomial Distribution 24
1.4 Important Continuous Distributions 27
1.4.1 The Normal Distribution 27
1.4.2 The Exponential Distribution 28
1.4.3 The Uniform Distribution 30
1.4.4 The Gamma Distribution 30
1.4.5 The Beta Distribution 31
1.4.6 The Joint Normal Distribution 31
1.5 Some Elementary Exercises 34
1.5.1 Tail Probabilities 34
1.5.2 The Exponential Distribution 37
1.6 Useful Functions, Integrals, and Sums 42
Contents
Conditional Probability and Conditional Expectation 47
2.1 The Discrete Case 47
2.2 The Dice Game Craps 52
2.3 Random Sums 57
2.3.1 Conditional Distributions: The Mixed Case 58
2.3.2 The Moments of a Random Sum 59
2.3.3 The Distribution of a Random Sum 61
2.4 Conditioning on a Continuous Random Variable 65
2.5 Martingales 71
2.5.1 The Definition 72
2.5.2 The Markov Inequality 73
2.5.3 The Maximal Inequality for Nonnegative Martingales 73
Markov Chains: Introduction 79
3.1 Definitions 79
3.2 Transition Probability Matrices of a Markov Chain 83
3.3 Some Markov Chain Models 87
3.3.1 An Inventory Model 87
3.3.2 The Ehrenfest Urn Model 89
3.3.3 Markov Chains in Genetics 90
3.3.4 A Discrete Queueing Markov Chain 92
3.4 First Step Analysis 95
3.4.1 Simple First Step Analyses 95
3.4.2 The General Absorbing Markov Chain 102
3.5 Some Special Markov Chains 111
3.5.1 The Two-State Markov Chain 112
3.5.2 Markov Chains Defined by Independent
Random Variables 114
3.5.3 One-Dimensional Random Walks 116
3.5.4 Success Runs 120
3.6 Functionals of Random Walks and Success Runs 124
3.6.1 The General Random Walk 128
3.6.2 Cash Management 132
3.6.3 The Success Runs Markov Chain 134
3.7 Another Look at First Step Analysis 139
3.8 Branching Processes 146
3.8.1 Examples of Branching Processes 147
3.8.2 The Mean and Variance of a Branching Process 148
3.8.3 Extinction Probabilities 149
3.9 Branching Processes and Generating Functions 152
3.9.1 Generating Functions and Extinction Probabilities 154
3.9.2 Probability Generating Functions and Sums of
Independent Random Variables 157
3.9.3 Multiple Branching Processes 159
Contents
The Long Run Behavior of Markov Chains 165
4.1 Regular Transition Probability Matrices 165
4.1.1 Doubly Stochastic Matrices 170
4.1.2 Interpretation of the Limiting Distribution 171
4.2 Examples 178
4.2.1 Including History in the State Description 178
4.2.2 Reliability and Redundancy 179
4.2.3 A Continuous Sampling Plan 181
4.2.4 Age Replacement Policies 183
4.2.5 Optimal Replacement Rules 185
4.3 The Classification of States 194
4.3.1 Irreducible Markov Chains 195
4.3.2 Periodicity of a Markov Chain 196
4.3.3 Recurrent and Transient States 198
4.4 The Basic Limit Theorem of Markov Chains 203
4.5 Reducible Markov Chains 215
Poisson Processes 223
5.1 The Poisson Distribution and the Poisson Process 223
5.1.1 The Poisson Distribution 223
5.1.2 The Poisson Process 225
5.1.3 Nonhomogeneous Processes 226
5.1.4 Cox Processes 227
5.2 The Law of Rare Events 232
5.2.1 The Law of Rare Events and the Poisson Process 234
5.2.2 Proof of Theorem 5.3 237
5.3 Distributions Associated with the Poisson Process 241
5.4 The Uniform Distribution and Poisson Processes 247
5.4.1 Shot Noise 253
5.4.2 Sum Quota Sampling 255
5.5 Spatial Poisson Processes 259
5.6 Compound and Marked Poisson Processes 264
5.6.1 Compound Poisson Processes 264
5.6.2 Marked Poisson Processes 267
Continuous Time Markov Chains 277
6.1 Pure Birth Processes 277
6.1.1 Postulates for the Poisson Process 277
6.1.2 Pure Birth Process 278
6.1.3 The Yule Process 282
6.2 Pure Death Processes 286
6.2.1 The Linear Death Process 287
6.2.2 Cable Failure Under Static Fatigue 290
Contents
6.3 Birth and Death Processes 295
6.3.1 Postulates 295
6.3.2 Sojourn Times 296
6.3.3 Differential Equations of Birth and Death Processes 299
6.4 The Limiting Behavior of Birth and Death Processes 304
6.5 Birth and Death Processes with Absorbing States 316
6.5.1 Probability of Absorption into State 0 316
6.5.2 Mean Time Until Absorption 318
6.6 Finite-State Continuous Time Markov Chains 327
6.7 A Poisson Process with a Markov Intensity 338
7 Renewal Phenomena 347
7.1 Definition of a Renewal Process and Related Concepts 347
7.2 Some Examples of Renewal Processes 353
7.2.1 Brief Sketches of Renewal Situations 353
7.2.2 Block Replacement 354
7.3 The Poisson Process Viewed as a Renewal Process 358
7.4 The Asymptotic Behavior of Renewal Processes 362
7.4.1 The Elementary Renewal Theorem 363
7.4.2 The Renewal Theorem for Continuous Lifetimes 365
7.4.3 The Asymptotic Distribution of N(t) 367
7.4.4 The Limiting Distribution of Age and Excess Life 368
7.5 Generalizations and Variations on Renewal Processes 371
7.5.1 Delayed Renewal Processes 371
7.5.2 Stationary Renewal Processes 372
7.5.3 Cumulative and Related Processes 372
7.6 Discrete Renewal Theory 379
7.6.1 The Discrete Renewal Theorem 383
7.6.2 Deterministic Population Growth with Age Distribution 384
8 Brownian Motion and Related Processes 391
8.1 Brownian Motion and Gaussian Processes 391
8.1.1 A Little History 391
8.1.2 The Brownian Motion Stochastic Process 392
8.1.3 The Central Limit Theorem and the Invariance Principle 396
8.1.4 Gaussian Processes 398
8.2 The Maximum Variable and the Reflection Principle 405
8.2.1 The Reflection Principle 406
8.2.2 The Time to First Reach a Level 407
8.2.3 The Zeros of Brownian Motion 408
8.3 Variations and Extensions 411
8.3.1 Reflected Brownian Motion 411
8.3.2 Absorbed Brownian Motion 412
8.3.3 The Brownian Bridge 414
8.3.4 Brownian Meander 416
Contents
8.4 Brownian Motion with Drift 419
8.4.1 The Gambler s Ruin Problem 420
8.4.2 Geometric Brownian Motion 424
8.5 The Ornstein-Uhlenbeck Process 432
8.5.1 A Second Approach to Physical Brownian Motion 434
8.5.2 The Position Process 437
8.5.3 The Long Run Behavior 439
8.5.4 Brownian Measure and Integration 441
9 Queueing Systems 447
9.1 Queueing Processes 447
9.1.1 The Queueing Formula L = A. W 448
9.1.2 A Sampling of Queueing Models 449
9.2 Poisson Arrivals, Exponential Service Times 451
9.2.1 The M/M/l System 452
9.2.2 The M/M/oo System 456
9.2.3 The M/M/s System 457
9.3 General Service Time Distributions 460
9.3.1 The M/G/l System 460
9.3.2 The M/G/oo System 465
9.4 Variations and Extensions 468
9.4.1 Systems with Balking 468
9.4.2 Variable Service Rates 469
9.4.3 A System with Feedback 470
9.4.4 A Two-Server Overflow Queue 470
9.4.5 Preemptive Priority Queues 472
9.5 Open Acyclic Queueing Networks 480
9.5.1 The Basic Theorem 480
9.5.2 Two Queues in Tandem 481
9.5.3 Open Acyclic Networks 482
9.5.4 Appendix: Time Reversibility 485
9.5.5 Proof of Theorem 9.1 487
9.6 General Open Networks 488
9.6.1 The General Open Network 492
10 Random Evolutions 495
10.1 Two-State Velocity Model 495
10.1.1 Two-State Random Evolution 498
10.1.2 The Telegraph Equation 500
10.1.3 Distribution Functions and Densities in the
Two-State Model 501
10.1.4 Passage Time Distributions 505
10.2 TV-State Random Evolution 507
10.2.1 Finite Markov Chains and Random Velocity Models 507
10.2.2 Constructive Approach of Random Velocity Models 507
Contents
10.2.3 Random Evolution Processes 508
10.2.4 Existence-Uniqueness of the First-Order
System (10.26) 509
10.2.5 Single Hyperbolic Equation 510
10.2.6 Spectral Properties of the Transition Matrix 512
10.2.7 Recurrence Properties of Random Evolution 515
10.3 Weak Law and Central Limit Theorem 516
10.4 Isotropic Transport in Higher Dimensions 521
10.4.1 The Rayleigh Problem of Random Flights 521
10.4.2 Three-Dimensional Rayleigh Model 523
11 Characteristic Functions and Their Applications 525
11.1 Definition of the Characteristic Function 525
11.1.1 Two Basic Properties of the Characteristic Function 526
11.2 Inversion Formulas for Characteristic Functions 527
11.2.1 Fourier Reciprocity/Local Non-Uniqueness 530
11.2.2 Fourier Inversion and Parseval s Identity 531
11.3 Inversion Formula for General Random Variables 532
11.4 The Continuity Theorem 533
11.4.1 Proof of the Continuity Theorem 534
11.5 Proof of the Central Limit Theorem 535
11.6 Stirling s Formula and Applications 536
11.6.1 Poisson Representation of n! 537
11.6.2 Proof of Stirling s Formula 538
11.7 Local deMoivre-Laplace Theorem 539
Further Reading 541
Answers to Exercises 543
Index 557
|
| any_adam_object | 1 |
| author | Pinsky, Mark A. 1940- Karlin, Samuel 1924-2007 |
| author_GND | (DE-588)108438791 (DE-588)118918672 |
| author_facet | Pinsky, Mark A. 1940- Karlin, Samuel 1924-2007 |
| author_role | aut aut |
| author_sort | Pinsky, Mark A. 1940- |
| author_variant | m a p ma map s k sk |
| building | Verbundindex |
| bvnumber | BV036967231 |
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| dewey-full | 519.23011 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.23011 |
| dewey-search | 519.23011 |
| dewey-sort | 3519.23011 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 4. ed. |
| format | Book |
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| indexdate | 2024-07-09T22:51:44Z |
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| isbn | 9780123814166 |
| language | English |
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| spelling | Pinsky, Mark A. 1940- Verfasser (DE-588)108438791 aut An introduction to stochastic modeling Mark A. Pinsky ; Samuel Karlin 4. ed. Amsterdam [u.a.] Elsevier [u.a.] 2011 XIV, 563 S. graph. Darst. 23 cm txt rdacontent n rdamedia nc rdacarrier Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 s Markov-Prozess (DE-588)4134948-9 s DE-604 Stochastischer Prozess (DE-588)4057630-9 s 1\p DE-604 Karlin, Samuel 1924-2007 Verfasser (DE-588)118918672 aut Früher u.d.T. Taylor, Howard M. An Introduction to stochastic modeling HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020882013&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 |
| spellingShingle | Pinsky, Mark A. 1940- Karlin, Samuel 1924-2007 An introduction to stochastic modeling Markov-Prozess (DE-588)4134948-9 gnd Stochastisches Modell (DE-588)4057633-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4134948-9 (DE-588)4057633-4 (DE-588)4057630-9 |
| title | An introduction to stochastic modeling |
| title_auth | An introduction to stochastic modeling |
| title_exact_search | An introduction to stochastic modeling |
| title_full | An introduction to stochastic modeling Mark A. Pinsky ; Samuel Karlin |
| title_fullStr | An introduction to stochastic modeling Mark A. Pinsky ; Samuel Karlin |
| title_full_unstemmed | An introduction to stochastic modeling Mark A. Pinsky ; Samuel Karlin |
| title_old | Taylor, Howard M. An Introduction to stochastic modeling |
| title_short | An introduction to stochastic modeling |
| title_sort | an introduction to stochastic modeling |
| topic | Markov-Prozess (DE-588)4134948-9 gnd Stochastisches Modell (DE-588)4057633-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Markov-Prozess Stochastisches Modell Stochastischer Prozess |
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