Introduction to probability models:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam u.a.
Academic Press
2007
|
| Ausgabe: | 9. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XVIII, 782 S. Ill. |
| ISBN: | 9780125980623 0125980620 9780123736352 0123736358 |
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| 245 | 1 | 0 | |a Introduction to probability models |c Sheldon M. Ross |
| 250 | |a 9. ed. | ||
| 264 | 1 | |a Amsterdam u.a. |b Academic Press |c 2007 | |
| 300 | |a XVIII, 782 S. |b Ill. | ||
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| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Probabilidades | |
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Datensatz im Suchindex
| _version_ | 1804135734128410624 |
|---|---|
| adam_text | Contents
Preface xiii
1. Introduction to Probability Theory 1
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 21
2. Random Variables 23
2.1. Random Variables 23
2.2. Discrete Random Variables 27
2.2.1. TheBenroirlii Random Variable 28
2.2.2. Trje Binomial Random Variable 29
2.2.3. Trie Geometric Random Variable 31
2.2.4. The Pbisfcon Random Variable 32
2.3. Continuous Random Variables 34
2.3.1. THe Uniform Random Variable 35
2.3.2. Exponential Ra«dom Variables 36
2.3.3. Gamma Random Variables 37
2.3.4. Normal Random Variables 37
VI Contents
2.4. Expectation of a Random Variable 38
2.4.1. The Discrete Case 38
2.4.2. The Continuous Case 41
2.4.3. Expectation of a Function of a Random Variable 43
2.5. Jointly Distributed Random Variables 47
2.5.1. Joint Distribution Functions 47
2.5.2. Independent Random Variables 51
2.5.3. Covariance and Variance of Sums of Random Variables 53
2.5.4. Joint Probability Distribution of Functions of Random
Variables 61
2.6. Moment Generating Functions 64
2.6.1. The Joint Distribution of the Sample Mean and Sample
Variance from a Normal Population 74
2.7. Limit Theorems 77
2.8. Stochastic Processes 83
Exercises 85
References 96
3. Conditional Probability and Conditional
Expectation 97
3.1. Introduction 97
3.2. The Discrete Case 97
3.3. The Continuous Case 102
3.4. Computing Expectations by Conditioning 105
3.4.1. Computing Variances by Conditioning 117
3.5. Computing Probabilities by Conditioning 120
3.6. Some Applications 137
3.6.1. A List Model 137
3.6.2. A Random Graph 139
3.6.3. Uniform Priors, Polya s Urn Model, and
Bose Einstein Statistics 147
3.6.4. Mean Time for Patterns 151
3.6.5. The ^ Record Values of Discrete Random Variables 155
3.7. An Identity for Compound Random Variables 158
3.7.1. Poisson Compounding Distribution 161
3.7.2. Binomial Compounding Distribution 163
3.7.3. A Compounding Distribution Related to the Negative
Binomial 164
Exercises 165
VI Contents
2.4. Expectation of a Random Variable 38
2.4.1. The Discrete Case 38
2.4.2. The Continuous Case 41
2.4.3. Expectation of a Function of a Random Variable 43
2.5. Jointly Distributed Random Variables 47
2.5.1. Joint Distribution Functions 47
2.5.2. Independent Random Variables 51
2.5.3. Covariance and Variance of Sums of Random Variables 53
2.5.4. Joint Probability Distribution of Functions of Random
Variables 61
2.6. Moment Generating Functions 64
2.6.1. The Joint Distribution of the Sample Mean and Sample
Variance from a Normal Population 74
2.7. Limit Theorems 77
2.8. Stochastic Processes 83
Exercises 85
References 96
3. Conditional Probability and Conditional
Expectation 97
3.1. Introduction 97
3.2. The Discrete Case 97
3.3. The Continuous Case 102
3.4. Computing Expectations by Conditioning 105
3.4.1. Computing Variances by Conditioning 117
3.5. Computing Probabilities by Conditioning 120
3.6. Some Applications 137
3.6.1. A List Model 137
3.6.2. A Random Graph 139
3.6.3. Uniform Priors, Polya s Urn Model, and
Bose Einstein Statistics 147
3.6.4. Mean Time for Patterns 151
3.6.5. The ^ Record Values of Discrete Random Variables 155
3.7. An Identity for Compound Random Variables 158
3.7.1. Poisson Compounding Distribution 161
3.7.2. Binomial Compounding Distribution 163
3.7.3. A Compounding Distribution Related to the Negative
Binomial 164
Exercises 165
viii Contents
Exercises 346
References 364
6. Continuous Time Markov Chains 365
6.1. Introduction 365
6.2. Continuous Time Markov Chains 366
6.3. Birth and Death Processes 368
6.4. The Transition Probability Function Pu(t) 375
6.5. Limiting Probabilities 384
6.6. Time Reversibility 392
6.7. Uniformization 401
6.8. Computing the Transition Probabilities 404
Exercises 407
References 415
7. Renewal Theory and Its Applications 417
7.1. Introduction 417
7.2. Distribution of N(t) 419
7.3. Limit Theorems and Their Applications 423
7.4. Renewal Reward Processes 433
7.5. Regenerative Processes 442
7.5.1. Alternating Renewal Processes 445
7.6. Semi Markov Processes 452
7.7. The Inspection Paradox 455
7.8. Computing the Renewal Function 458
7.9. Applications to Patterns 461
7.9.1. Patterns of Discrete Random Variables 462
7.9.2. The Expected Time to a Maximal Run of Distinct Values 469
7.9.3. Increasing Runs of Continuous Random Variables 471
7.10. The Insurance Ruin Problem 473
Exercises 479
References 492
8. Queueing Theory 493
8.1. Introduction 493
8.2. Preliminaries 494
8.2.1. Cost Equations 495
8.2.2. Steady State Probabilities 496
Contents ix
8.3. Exponential Models 499
8.3.1. A Single Server Exponential Queueing System 499
8.3.2. A Single Server Exponential Queueing System
Having Finite Capacity 508
8.3.3. A Shoeshine Shop 511
8.3.4. A Queueing System with Bulk Service 514
8.4. Network of Queues 517
8.4.1. Open Systems 517
8.4.2. Closed Systems 522
8.5. The System M/G/l 528
8.5.1. Preliminaries: Work and Another Cost Identity 528
8.5.2. Application of Work to M/G/l 529
8.5.3. Busy Periods 530
8.6. Variations on the M/G/l 531
8.6.1. The M/G/1 with Random Sized Batch Arrivals 531
8.6.2. Priority Queues 533
8.6.3. An M/G/l Optimization Example 536
8.6.4. The M/G/l Queue with Server Breakdown 540
8.7. The Model G/M/l 543
8.7.1. The G/M/1 Busy and Idle Periods 548
8.8. A Finite Source Model 549
8.9. Multiserver Queues 552
8.9.1. Erlang s Loss System 553
8.9.2. The A//M/£ Queue 554
8.9.3. The G/M/k Queue 554
8.9.4. The M/G/k Queue 556
Exercises 558
References 570
9. Reliability Theory 571
9.1. Introduction 571
9.2. Structure Functions 571
9.2.1. Minimal Path and Minimal Cut Sets 574
9.3. Reliability of Systems of Independent Components 578
9.4. Bounds on the Reliability Function 583
9.4.1. Method of Inclusion and Exclusion 584
9.4.2. Second Method for Obtaining Bounds on r(p) 593
9.5. System Life as a Function of Component Lives 595
9.6. Expected System Lifetime 604
9.6.1. An Upper Bound on the Expected Life of a
Parallel System 608
X Contents
9.7. Systems with Repair 610
9.7.1. A Series Model with Suspended Animation 615
Exercises 617
References 624
10. Brownian Motion and Stationary Processes 625
10.1. Brownian Motion 625
10.2. Hitting Times, Maximum Variable, and the Gambler s Ruin
Problem 629
10.3. Variations on Brownian Motion 631
10.3.1. Brownian Motion with Drift 631
10.3.2. Geometric Brownian Motion 631
10.4. Pricing Stock Options 632
10.4.1. An Example in Options Pricing 632
10.4.2. The Arbitrage Theorem 635
10.4.3. The Black Scholes Option Pricing Formula 638
10.5. White Noise 644
10.6. Gaussian Processes 646
10.7. Stationary and Weakly Stationary Processes 649
10.8. Harmonic Analysis of Weakly Stationary Processes 654
Exercises 657
References 662
11. Simulation 663
11.1. Introduction 663
11.2. General Techniques for Simulating Continuous Random
Variables 668
11.2.1. The Inverse Transformation Method 668
11.2.2. The Rejection Method 669
11.2.3. The Hazard Rate Method 673
11.3. Special Techniques for Simulating Continuous Random
Variables 677
11.3.1. The Normal Distribution 677
11.3.2. The Gamma Distribution 680
11.3.3. The Chi Squared Distribution 681
11.3.4. The Beta (n, m) Distribution 681
11.3.5. The Exponential Distribution—The Von Neumann
Algorithm 682
11.4. Simulating from Discrete Distributions 685
11.4.1. The Alias Method 688
Contents XI
11.5. Stochastic Processes 692
11.5.1. Simulating a Nonhomogeneous Poisson Process 693
11.5.2. Simulating a Two Dimensional Poisson Process 700
11.6. Variance Reduction Techniques 703
11.6.1. Use of Antithetic Variables 704
11.6.2. Variance Reduction by Conditioning 708
11.6.3. Control Variates 712
11.6.4. Importance Sampling 714
11.7. Determining the Number of Runs 720
11.8. Coupling from the Past 720
Exercises 723
References 731
Appendix: Solutions to Starred Exercises 733
Index 775
|
| adam_txt |
Contents
Preface xiii
1. Introduction to Probability Theory 1
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 21
2. Random Variables 23
2.1. Random Variables 23
2.2. Discrete Random Variables 27
2.2.1. TheBenroirlii Random Variable 28
2.2.2. Trje Binomial Random Variable 29
2.2.3. Trie Geometric Random Variable 31
2.2.4. The Pbisfcon Random Variable 32
2.3. Continuous Random Variables 34
2.3.1. THe Uniform Random Variable 35
2.3.2. Exponential Ra«dom Variables 36
2.3.3. Gamma Random Variables 37
2.3.4. Normal Random Variables 37
VI Contents
2.4. Expectation of a Random Variable 38
2.4.1. The Discrete Case 38
2.4.2. The Continuous Case 41
2.4.3. Expectation of a Function of a Random Variable 43
2.5. Jointly Distributed Random Variables 47
2.5.1. Joint Distribution Functions 47
2.5.2. Independent Random Variables 51
2.5.3. Covariance and Variance of Sums of Random Variables 53
2.5.4. Joint Probability Distribution of Functions of Random
Variables 61
2.6. Moment Generating Functions 64
2.6.1. The Joint Distribution of the Sample Mean and Sample
Variance from a Normal Population 74
2.7. Limit Theorems 77
2.8. Stochastic Processes 83
Exercises 85
References 96
3. Conditional Probability and Conditional
Expectation 97
3.1. Introduction 97
3.2. The Discrete Case 97
3.3. The Continuous Case 102
3.4. Computing Expectations by Conditioning 105
3.4.1. Computing Variances by Conditioning 117
3.5. Computing Probabilities by Conditioning 120
3.6. Some Applications 137
3.6.1. A List Model 137
3.6.2. A Random Graph 139
3.6.3. Uniform Priors, Polya's Urn Model, and
Bose Einstein Statistics 147
3.6.4. Mean Time for Patterns 151
3.6.5. The ^ Record Values of Discrete Random Variables 155
3.7. An Identity for Compound Random Variables 158
3.7.1. Poisson Compounding Distribution 161
3.7.2. Binomial Compounding Distribution 163
3.7.3. A Compounding Distribution Related to the Negative
Binomial 164
Exercises 165
VI Contents
2.4. Expectation of a Random Variable 38
2.4.1. The Discrete Case 38
2.4.2. The Continuous Case 41
2.4.3. Expectation of a Function of a Random Variable 43
2.5. Jointly Distributed Random Variables 47
2.5.1. Joint Distribution Functions 47
2.5.2. Independent Random Variables 51
2.5.3. Covariance and Variance of Sums of Random Variables 53
2.5.4. Joint Probability Distribution of Functions of Random
Variables 61
2.6. Moment Generating Functions 64
2.6.1. The Joint Distribution of the Sample Mean and Sample
Variance from a Normal Population 74
2.7. Limit Theorems 77
2.8. Stochastic Processes 83
Exercises 85
References 96
3. Conditional Probability and Conditional
Expectation 97
3.1. Introduction 97
3.2. The Discrete Case 97
3.3. The Continuous Case 102
3.4. Computing Expectations by Conditioning 105
3.4.1. Computing Variances by Conditioning 117
3.5. Computing Probabilities by Conditioning 120
3.6. Some Applications 137
3.6.1. A List Model 137
3.6.2. A Random Graph 139
3.6.3. Uniform Priors, Polya's Urn Model, and
Bose Einstein Statistics 147
3.6.4. Mean Time for Patterns 151
3.6.5. The ^ Record Values of Discrete Random Variables 155
3.7. An Identity for Compound Random Variables 158
3.7.1. Poisson Compounding Distribution 161
3.7.2. Binomial Compounding Distribution 163
3.7.3. A Compounding Distribution Related to the Negative
Binomial 164
Exercises 165
viii Contents
Exercises 346
References 364
6. Continuous Time Markov Chains 365
6.1. Introduction 365
6.2. Continuous Time Markov Chains 366
6.3. Birth and Death Processes 368
6.4. The Transition Probability Function Pu(t) 375
6.5. Limiting Probabilities 384
6.6. Time Reversibility 392
6.7. Uniformization 401
6.8. Computing the Transition Probabilities 404
Exercises 407
References 415
7. Renewal Theory and Its Applications 417
7.1. Introduction 417
7.2. Distribution of N(t) 419
7.3. Limit Theorems and Their Applications 423
7.4. Renewal Reward Processes 433
7.5. Regenerative Processes 442
7.5.1. Alternating Renewal Processes 445
7.6. Semi Markov Processes 452
7.7. The Inspection Paradox 455
7.8. Computing the Renewal Function 458
7.9. Applications to Patterns 461
7.9.1. Patterns of Discrete Random Variables 462
7.9.2. The Expected Time to a Maximal Run of Distinct Values 469
7.9.3. Increasing Runs of Continuous Random Variables 471
7.10. The Insurance Ruin Problem 473
Exercises 479
References 492
8. Queueing Theory 493
8.1. Introduction 493
8.2. Preliminaries 494
8.2.1. Cost Equations 495
8.2.2. Steady State Probabilities 496
Contents ix
8.3. Exponential Models 499
8.3.1. A Single Server Exponential Queueing System 499
8.3.2. A Single Server Exponential Queueing System
Having Finite Capacity 508
8.3.3. A Shoeshine Shop 511
8.3.4. A Queueing System with Bulk Service 514
8.4. Network of Queues 517
8.4.1. Open Systems 517
8.4.2. Closed Systems 522
8.5. The System M/G/l 528
8.5.1. Preliminaries: Work and Another Cost Identity 528
8.5.2. Application of Work to M/G/l 529
8.5.3. Busy Periods 530
8.6. Variations on the M/G/l 531
8.6.1. The M/G/1 with Random Sized Batch Arrivals 531
8.6.2. Priority Queues 533
8.6.3. An M/G/l Optimization Example 536
8.6.4. The M/G/l Queue with Server Breakdown 540
8.7. The Model G/M/l 543
8.7.1. The G/M/1 Busy and Idle Periods 548
8.8. A Finite Source Model 549
8.9. Multiserver Queues 552
8.9.1. Erlang's Loss System 553
8.9.2. The A//M/£ Queue 554
8.9.3. The G/M/k Queue 554
8.9.4. The M/G/k Queue 556
Exercises 558
References 570
9. Reliability Theory 571
9.1. Introduction 571
9.2. Structure Functions 571
9.2.1. Minimal Path and Minimal Cut Sets 574
9.3. Reliability of Systems of Independent Components 578
9.4. Bounds on the Reliability Function 583
9.4.1. Method of Inclusion and Exclusion 584
9.4.2. Second Method for Obtaining Bounds on r(p) 593
9.5. System Life as a Function of Component Lives 595
9.6. Expected System Lifetime 604
9.6.1. An Upper Bound on the Expected Life of a
Parallel System 608
X Contents
9.7. Systems with Repair 610
9.7.1. A Series Model with Suspended Animation 615
Exercises 617
References 624
10. Brownian Motion and Stationary Processes 625
10.1. Brownian Motion 625
10.2. Hitting Times, Maximum Variable, and the Gambler's Ruin
Problem 629
10.3. Variations on Brownian Motion 631
10.3.1. Brownian Motion with Drift 631
10.3.2. Geometric Brownian Motion 631
10.4. Pricing Stock Options 632
10.4.1. An Example in Options Pricing 632
10.4.2. The Arbitrage Theorem 635
10.4.3. The Black Scholes Option Pricing Formula 638
10.5. White Noise 644
10.6. Gaussian Processes 646
10.7. Stationary and Weakly Stationary Processes 649
10.8. Harmonic Analysis of Weakly Stationary Processes 654
Exercises 657
References 662
11. Simulation 663
11.1. Introduction 663
11.2. General Techniques for Simulating Continuous Random
Variables 668
11.2.1. The Inverse Transformation Method 668
11.2.2. The Rejection Method 669
11.2.3. The Hazard Rate Method 673
11.3. Special Techniques for Simulating Continuous Random
Variables 677
11.3.1. The Normal Distribution 677
11.3.2. The Gamma Distribution 680
11.3.3. The Chi Squared Distribution 681
11.3.4. The Beta (n, m) Distribution 681
11.3.5. The Exponential Distribution—The Von Neumann
Algorithm 682
11.4. Simulating from Discrete Distributions 685
11.4.1. The Alias Method 688
Contents XI
11.5. Stochastic Processes 692
11.5.1. Simulating a Nonhomogeneous Poisson Process 693
11.5.2. Simulating a Two Dimensional Poisson Process 700
11.6. Variance Reduction Techniques 703
11.6.1. Use of Antithetic Variables 704
11.6.2. Variance Reduction by Conditioning 708
11.6.3. Control Variates 712
11.6.4. Importance Sampling 714
11.7. Determining the Number of Runs 720
11.8. Coupling from the Past 720
Exercises 723
References 731
Appendix: Solutions to Starred Exercises 733
Index 775 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Ross, Sheldon M. 1943- |
| author_GND | (DE-588)123762235 |
| author_facet | Ross, Sheldon M. 1943- |
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| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
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| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| edition | 9. ed. |
| format | Book |
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| id | DE-604.BV021812561 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:51:27Z |
| indexdate | 2024-07-09T20:45:12Z |
| institution | BVB |
| isbn | 9780125980623 0125980620 9780123736352 0123736358 |
| language | English |
| lccn | 2006051040 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015024823 |
| oclc_num | 71552484 |
| open_access_boolean | |
| owner | DE-384 DE-945 DE-703 DE-M347 DE-92 DE-526 DE-91 DE-BY-TUM DE-11 |
| owner_facet | DE-384 DE-945 DE-703 DE-M347 DE-92 DE-526 DE-91 DE-BY-TUM DE-11 |
| physical | XVIII, 782 S. Ill. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Academic Press |
| record_format | marc |
| spelling | Ross, Sheldon M. 1943- Verfasser (DE-588)123762235 aut Introduction to probability models Sheldon M. Ross 9. ed. Amsterdam u.a. Academic Press 2007 XVIII, 782 S. Ill. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Probabilidades Toepassingen gtt Waarschijnlijkheidstheorie gtt Probabilities Bodenchemie (DE-588)4146134-4 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Ökologische Chemie (DE-588)4135167-8 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 Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Mathematisches Modell (DE-588)4114528-8 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 Modell (DE-588)4039798-1 s 2\p DE-604 3\p DE-604 Bodenchemie (DE-588)4146134-4 s Ökologische Chemie (DE-588)4135167-8 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015024823&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 Probabilidades Toepassingen gtt Waarschijnlijkheidstheorie gtt Probabilities Bodenchemie (DE-588)4146134-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Ökologische Chemie (DE-588)4135167-8 gnd Modell (DE-588)4039798-1 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd Stochastisches Modell (DE-588)4057633-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4146134-4 (DE-588)4057630-9 (DE-588)4135167-8 (DE-588)4039798-1 (DE-588)4064324-4 (DE-588)4114528-8 (DE-588)4057633-4 (DE-588)4079013-7 |
| title | Introduction to probability models |
| title_auth | Introduction to probability models |
| title_exact_search | Introduction to probability models |
| title_exact_search_txtP | 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 | Probabilidades Toepassingen gtt Waarschijnlijkheidstheorie gtt Probabilities Bodenchemie (DE-588)4146134-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Ökologische Chemie (DE-588)4135167-8 gnd Modell (DE-588)4039798-1 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd Stochastisches Modell (DE-588)4057633-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Probabilidades Toepassingen Waarschijnlijkheidstheorie Probabilities Bodenchemie Stochastischer Prozess Ökologische Chemie Modell Wahrscheinlichkeitsrechnung Mathematisches Modell Stochastisches Modell Wahrscheinlichkeitstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015024823&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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