Introduction to probability models:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | Undetermined |
| Published: |
Boston [u. a.]
Acad. Pr.
1993
|
| Edition: | 5. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 556 S. |
| ISBN: | 0125984553 |
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| 245 | 1 | 0 | |a Introduction to probability models |c Sheldon M. Ross |
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| adam_text | IMAGE 1
CONTENTS
PREFACE XI
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
IMAGE 2
2.5. JOINTLY DISTRIBUTED RANDOM VARIABLES 44
2.5.1. JOINT DISTRIBUTION FUNCTIONS 44
2.5.2. INDEPENDENT RANDOM VARIABLES 48
2.5.3. JOINT PROBABILITY DISTRIBUTION OF FUNCTIONS OF RANDOM VARIABLES
55
2.6. MOMENT GENERATING FUNCTIONS 58
2.7. LIMIT THEOREMS 66
2.8. STOCHASTIC PROCESSES 70
EXERCISES 72
REFERENCES 82
3. CONDITIONAL PROBABILITY AND CONDITIONAL EXPECTATION
3.1. INTRODUCTION 83
3.2. THE DISCRETE CASE 83
3.3. THE CONTINUOUS CASE 88
3.4. COMPUTING EXPECTATIONS BY CONDITIONING 91
3.5. COMPUTING PROBABILITIES BY CONDITIONING 100
3.6. SOME APPLICATIONS 107
3.6.1. A LIST MODEL 107
3.6.2. A RANDOM GRAPH 108
3.6.3. UNIFORM PRIORS, POLYA S URN MODEL, AND BOSE-EINSTEIN STATISTICS
116
3.6.4. IN NORMAL SAMPLING X AND S 2 ARE INDEPENDENT 120
EXERCISES 125
4. MARKOV CHAINS
4.1. INTRODUCTION 137
4.2. CHAPMAN-KOLMOGOROV EQUATIONS 140
4.3. CLASSIFICATION OF STATES 143
4.4. LIMITING PROBABILITIES 151
4.5. SOME APPLICATIONS 161
4.5.1. THE GAMBLER S RUIN PROBLEM 161
4.5.2. A MODEL FOR ALGORITHMIC EFFICIENCY 165
4.6. BRANCHING PROCESSES 168
4.7. TIME REVERSIBLE MARKOV CHAINS 171
4.8. MARKOV DECISION PROCESSES 182
EXERCISES 186
REFERENCES 198
IMAGE 3
5. THE EXPONENTIAL DISTRIBUTION AND THE POISSON PROCESS
5.1. INTRODUCTION 199
5.2. THE EXPONENTIAL DISTRIBUTION 200
5.2.1. DEFINITION 200
5.2.2. PROPERTIES OF THE EXPONENTIAL DISTRIBUTION 201
5.2.3. FURTHER PROPERTIES OF THE EXPONENTIAL DISTRIBUTION 205 5.3. THE
POISSON PROCESS 208
5.3.1. COUNTING PROCESSES 208
5.3.2. DEFINITION OF THE POISSON PROCESS 209
5.3.3. INTERARRIVAL AND WAITING TIME DISTRIBUTIONS 214 5.3.4. FURTHER
PROPERTIES OF POISSON PROCESSES 216
5.3.5. CONDITIONAL DISTRIBUTION OF THE ARRIVAL TIMES 222 5.3.6.
ESTIMATING SOFTWARE RELIABILITY 233
5.4. GENERALIZATIONS OF THE POISSON PROCESS 235
5.4.1. NONHOMOGENEOUS POISSON PROCESS 235
5.4.2. COMPOUND POISSON PROCESS 239
EXERCISES 243
REFERENCES 254
6. CONTINUOUS-TIME MARKOV CHAINS
6.1. INTRODUCTION 255
6.2. CONTINUOUS-TIME MARKOV CHAINS 256
6.3. BIRTH AND DEATH PROCESSES 258
6.4. THE KOLMOGOROV DIFFERENTIAL EQUATIONS 265
6.5. LIMITING PROBABILITIES 272
6.6. TIME REVERSIBILITY 280
6.7. UNIFORMIZATION 286
6.8. COMPUTING THE TRANSITION PROBABILITIES 289
EXERCISES 292
REFERENCES 301
7. RENEWAL THEORY AND ITS APPLICATIONS
7.1. INTRODUCTION 303
7.2. DISTRIBUTION OF N(T) 305
7.3. LIMIT THEOREMS AND THEIR APPLICATIONS 309
7.4. RENEWAL REWARD PROCESSES 318
IMAGE 4
7.5. REGENERATIVE PROCESSES 325
7.5.1. ALTERNATING RENEWAL PROCESSES 326
7.6. SEMI-MARKOV PROCESSES 331
7.7. THE INSPECTION PARADOX 334
7.8. COMPUTING THE RENEWAL FUNCTION 336
EXERCISES 339
REFERENCES 349
8. QUEUEING THEORY
8.1. INTRODUCTION 351
8.2. PRELIMINARIES 352
8.2.1. COST EQUATIONS 352
8.2.2. STEADY-STATE PROBABILITIES 354
8.3. EXPONENTIAL MODELS 356
8.3.1. A SINGLE-SERVER EXPONENTIAL QUEUEING SYSTEM 356 8.3.2. A
SINGLE-SERVER EXPONENTIAL QUEUEING SYSTEM HAVING FINITE CAPACITY 363
8.3.3. A SHOESHINE SHOP 366
8.3.4. A QUEUEING SYSTEM WITH BULK SERVICE 369
8.4. NETWORK OF QUEUES 372
8.4.1. OPEN SYSTEMS 372
8.4.2. CLOSED SYSTEMS 377
8.5. THE SYSTEM M/G/L 381
8.5.1. PRELIMINARIES: WORK AND ANOTHER COST IDENTITY 381 8.5.2.
APPLICATION OF WORK TO M/G/L 382
8.5.3. BUSY PERIODS 383
8.6. VARIATIONS ON THE M/G/L 385
8.6.1. THE M/G/L WITH RANDOM-SIZED BATCH ARRIVALS 385 8.6.2. PRIORITY
QUEUES 387
8.7. THE MODEL G/M/L 390
8.7.1. THE G/M/L BUSY AND IDLE PERIODS 394
8.8. MULTISERVER QUEUES 395
8.8.1. ERLANG S LOSS SYSTEM 395
8.8.2. THE M/M/K QUEUE 397
8.8.3. THE G/M/K QUEUE 397
8.8.4. THE M/G/K QUEUE 399
EXERCISES 401
REFERENCES 410
IMAGE 5
9. RELIABILITY THEORY
9.1. INTRODUCTION 411
9.2. STRUCTURE FUNCTIONS 412
9.2.1. MINIMAL PATH AND MINIMAL CUT SETS 414
9.3. RELIABILITY OF SYSTEMS OF INDEPENDENT COMPONENTS 418 9.4. BOUNDS ON
THE RELIABILITY FUNCTION 422
9.4.1. METHOD OF INCLUSION AND EXCLUSION 423
9.4.2. SECOND METHOD FOR OBTAINING BOUNDS ON R(P) 431 9.5. SYSTEM LIFE
AS A FUNCTION OF COMPONENT LIVES 433
9.6. EXPECTED SYSTEM LIFETIME 441
9.7. SYSTEMS WITH REPAIR 445
EXERCISES 449
REFERENCES 456
10. BROWNIAN MOTION AND STATIONARY PROCESSES
10.1. BROWNIAN MOTION 457
10.2. HITTING TIMES, MAXIMUM VARIABLE, AND THE GAMBLER S RUIN PROBLEM
460
10.3. VARIATIONS ON BROWNIAN MOTION 462
10.3.1. BROWNIAN MOTION WITH DRIFT 462
10.3.2. GEOMETRIC BROWNIAN MOTION 462
10.4. PRICING STOCK OPTIONS 463
10.4.1. AN EXAMPLE IN OPTIONS PRICING 463
10.4.2. THE ARBITRAGE THEOREM 466
10.4.3. THE BLACK-SCHOLES OPTION PRICING FORMULA 469 10.5. WHITE NOISE
474
10.6. GAUSSIAN PROCESSES 476
10.7. STATIONARY AND WEAKLY STATIONARY PROCESSES 479
10.8. HARMONIC ANALYSIS OF WEAKLY STATIONARY PROCESSES 484 EXERCISES 486
REFERENCES 491
11. SIMULATION
11.1. INTRODUCTION 493
11.2. GENERAL TECHNIQUES FOR SIMULATING CONTINUOUS RANDOM VARIABLES 498
IMAGE 6
11.2.1. THE INVERSE TRANSFORMATION METHOD 498
11.2.2. THE REJECTION METHOD 499
11.2.3. HAZARD RATE METHOD 503
11.3. SPECIAL TECHNIQUES FOR SIMULATING CONTINUOUS RANDOM VARIABLES 506
11.3.1. THE NORMAL DISTRIBUTION 506
11.3.2. THE GAMMA DISTRIBUTION 510
11.3.3. THE CHI-SQUARED DISTRIBUTION 510
11.3.4. THE BETA (N, M) DISTRIBUTION 511
11.3.5. THE EXPONENTIAL DISTRIBUTION-THE VON NEUMANN ALGORITHM 512
11.4. SIMULATING FROM DISCRETE DISTRIBUTIONS 514
11.4.1. THE ALIAS METHOD 518
11.5. STOCHASTIC PROCESSES 521
11.5.1. SIMULATING A NONHOMOGENEOUS POISSON PROCESS 523 11.5.2.
SIMULATING A TWO-DIMENSIONAL POISSON PROCESS 529 11.6. VARIANCE
REDUCTION TECHNIQUES 532
11.6.1. USE OF ANTITHETIC VARIABLES 533
11.6.2. VARIANCE REDUCTION BY CONDITIONING 536
11.6.3. CONTROL VARIATES 540
11.7. DETERMINING THE NUMBER OF RUNS 542
EXERCISES 542
REFERENCES 551
INDEX 553
|
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| discipline | Mathematik |
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| spelling | Ross, Sheldon M. 1943- Verfasser (DE-588)123762235 aut Introduction to probability models Sheldon M. Ross 5. ed. Boston [u. a.] Acad. Pr. 1993 XI, 556 S. txt rdacontent n rdamedia nc rdacarrier Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Modell (DE-588)4039798-1 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 s Stochastisches Modell (DE-588)4057633-4 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Modell (DE-588)4039798-1 s Mathematisches Modell (DE-588)4114528-8 s SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019130798&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ross, Sheldon M. 1943- Introduction to probability models Mathematisches Modell (DE-588)4114528-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Modell (DE-588)4039798-1 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| subject_GND | (DE-588)4114528-8 (DE-588)4064324-4 (DE-588)4039798-1 (DE-588)4079013-7 (DE-588)4057630-9 (DE-588)4057633-4 |
| 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 | Mathematisches Modell (DE-588)4114528-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Modell (DE-588)4039798-1 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| topic_facet | Mathematisches Modell Wahrscheinlichkeitsrechnung Modell Wahrscheinlichkeitstheorie Stochastischer Prozess Stochastisches Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019130798&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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