Theoretical probability for applications:
Offering comprehensive coverage of modern probability theory (exclusive of continuous time stochastic processes), this unique book functions as both an introduction for graduate statisticians, mathematicians, engineers, and economists and an encyclopedic reference of the subject for professionals in...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Wiley
1994
|
| Schriftenreihe: | Wiley series in probability and mathematical statistics
A Wiley interscience publication |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | Offering comprehensive coverage of modern probability theory (exclusive of continuous time stochastic processes), this unique book functions as both an introduction for graduate statisticians, mathematicians, engineers, and economists and an encyclopedic reference of the subject for professionals in these fields. It assumes only a knowledge of calculus as well as basic real analysis and linear algebra Throughout Theoretical Probability for Applications the focus is on the practical uses of this increasingly important tool. It develops topics of discrete time probability theory for use in a multitude of applications, including stochastic processes, theoretical statistics, and other disciplines that require a sound foundation in modern probability theory. Principles of measure theory related to the study of probability theory are developed as they are required throughout the book The book examines most of the basic probability models that involve only a finite or countably infinite number of random variables. Topics in the "Discrete Models" section include Bernoulli trials, random walks, matching, sums of indicators, multinomial trials. Poisson approximations and processes, sampling. Markov chains, and discrete renewal theory. Nondiscrete models discussed include univariate, Beta, sampling, and Dirichlet distributions as well as order statistics |
| Beschreibung: | XVIII, 894 S. |
| ISBN: | 0471632163 |
Internformat
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| 300 | |a XVIII, 894 S. | ||
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| 490 | 0 | |a Wiley series in probability and mathematical statistics | |
| 490 | 0 | |a A Wiley interscience publication | |
| 520 | 3 | |a Offering comprehensive coverage of modern probability theory (exclusive of continuous time stochastic processes), this unique book functions as both an introduction for graduate statisticians, mathematicians, engineers, and economists and an encyclopedic reference of the subject for professionals in these fields. It assumes only a knowledge of calculus as well as basic real analysis and linear algebra | |
| 520 | |a Throughout Theoretical Probability for Applications the focus is on the practical uses of this increasingly important tool. It develops topics of discrete time probability theory for use in a multitude of applications, including stochastic processes, theoretical statistics, and other disciplines that require a sound foundation in modern probability theory. Principles of measure theory related to the study of probability theory are developed as they are required throughout the book | ||
| 520 | |a The book examines most of the basic probability models that involve only a finite or countably infinite number of random variables. Topics in the "Discrete Models" section include Bernoulli trials, random walks, matching, sums of indicators, multinomial trials. Poisson approximations and processes, sampling. Markov chains, and discrete renewal theory. Nondiscrete models discussed include univariate, Beta, sampling, and Dirichlet distributions as well as order statistics | ||
| 650 | 7 | |a Probabilités |2 ram | |
| 650 | 7 | |a Waarschijnlijkheidstheorie |2 gtt | |
| 650 | 4 | |a Probabilities | |
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Datensatz im Suchindex
| _version_ | 1804123528204648448 |
|---|---|
| adam_text | Contents
PART I FUNDAMENTALS OF PROBABILITY THEORY 1
1. Random Phenomena 3
2. Experiments 5
3. Finite Probability Spaces 7
3.1. Set Operations, 8
Exercises, 10
4. Counting Rules 12
4.1. Poker Hands, 15
Exercises, 18
5. Probability Spaces 19
5.1. Discrete Probability Spaces, 19
5.2. An Experiment with Uncountably Many Outcomes, 21
5.3. Measurable Spaces, 22
5.3.1. The ff Field Generated by %, 24
5.4. General Probability Spaces, 25
5.4.1. Rules of Probability, 25
5.5. Borel Sets, 28
5.6. Determining Probability Measures, 30
5.7. Construction of Probability Measures, 31
Exercises, 34
6. Conditional Probability 40
Exercises, 43
vii
viii CONTENTS
7. Independence 46
7.1. Compound Experiments, 49
7.2. Independence of Collections, 50
8. Probabilities on Euclidean Spaces 52
8.1. Introduction, 52
8.2. Discrete Distributions, 53
8.3. Lebesgue Integration, 54
8.3.1. Sets of Measure 0, 54
8.3.2. Definition of the Lebesgue Integral, 57
8.3.3. Properties of the Lebesgue Integral, 60
8.3.4. Riemann Integration vs. Lebesgue Integration, 63
8.3.5. Lebesgue Measurable Functions, 65
8.3.6. Fubini s Theorem, 68
8.4. Absolutely Continuous Distributions, 70
8.5. Regularity Properties of Distributions on Rd, 72
Exercises, 74
9. Measurable Functions and Random Vectors 75
9.1. Measurable Transformations, 76
9.2. Properties of Random Vectors, 78
9.3. Extended Random Variables, 80
9.4. Distributions of Measurable Mappings, 83
9.5. Marginal and Joint Distributions, 85
9.6. Independent Random Vectors, 87
9.7. Product Spaces, 90
9.8. Mixed Distributions, 93
Exercises, 93
10. Distribution Functions 95
10.1. One Dimensional Distribution Functions, 95
10.2. Inverse Distribution Function, 98
10.3. Distribution Functions on Rd, 101
10.4. Extended Distribution Functions, 107
Exercises, 108
CONTENTS ix
11. Expectation 109
11.1. Expectation for Simple Random Variables, 109
11.2. Expectation for Nonnegative Random Variables, 110
11.3. Expectation in General, 117
11.4. Expectation of Functions of a Random Vector, 120
11.5. Expectation of a Random Vector, 124
11.6. Expectation as an Integral, 125
11.7. Product Spaces, 127
11.8. Limits and Expectation, 135
11.9. Moments, 138
Exercises, 140
12. Some Inequalities 147
13. Covariance 151
13.1. Covariance in R, 151
13.2. Covariance in Rd, 155
13.3. Approximations for the Mean and Variance, 159
14. Conditioning 162
14.1. Conditioning on an Event, 162
14.2. Conditioning on a Discrete Random Vector, 163
14.3. Conditioning on a c Field, 166
14.4. Properties of Conditional Expectation, 169
14.5. Optimal Predictors, 177
14.6. Regular Conditional Probabilities, 178
14.7. Kernels and Regular Conditional Probabilities, 183
14.7.1. Conditional Distributions Given X, 184
14.7.2. Probabilities on Product Spaces, 186
Exercises, 192
15. Exchangeability 197
16. Infinite Sequences of Random Vectors 202
16.1. Distribution of a Random Sequence, 202
16.2. Infinite Product Measure, 204
16.3. Measurability, 206
16.4. Conditioning, 207
X CONTENTS
16.5. Stochastic Processes, 208
16.6 Kolmogorov s Existence Theorem, 210
17. Introduction to Martingales 213
17.1. Transformations and Gambling Systems, 214
17.2. Stopping, 216
17.3. Maximal Inequality, 220
17.4. Relative Submartingales, 221
Exercises, 225
PART II DISCRETE MODELS 231
18. Introduction to Parts II IV 233
19. Success Failure Experiments 234
Exercises, 237
20. Generating Functions 238
Exercises, 243
21. Waiting Times in Bernoulli Trials 246
21.1. Geometric Distributions, 246
21.2. Negative Binomial Distributions, 248
21.3. Joint Distributions, 251
21.4. Collector s Problem, 252
Exercises, 253
22. Conditional Distributions in Bernoulli Trials 258
23. Random Bernoulli Trials 261
23.1. A Bayesian Viewpoint, 263
Exercises, 265
24. Blood Tests 266
Exercises, 268
CONTENTS XI
25. Random Walks 269
25.1. Unrestricted Walk, 269
25.2. The Gambler s Ruin, 269
25.3. First Passage Probability, 273
25.4. Use of Martingales, 274
25.5. First Passage Times, 277
Exercises, 280
26. Matching 283
26.1. Completely Random Matching, 283
26.2. Matching Strategies, 284
Exercises, 286
27. Sums of Indicators 288
Exercises, 293
28. Multinomial Trials 297
28.1. Distribution of the Occupancy Vector, 297
28.2. Classical Occupancy, 301
28.3. Random Multinomial Trials, 302
Exercises, 304
29. Poisson Approximations and Poisson Processes 308
29.1. Poisson Approximations, 308
29.2. Poisson Processes, 310
29.3. Examples, 312
29.4. Construction of a Poisson Process, 313
29.5. A Characterization of the Poisson Processes, 316
29.6. Distance to the Particles, 317
29.7. Other Characterizations of a Poisson Process, 318
29.8. Compound Poisson Processes, 321
29.9. Point Processes, 323
Exercises, 324
Xli CONTENTS
30. Stopping Multinomial Trials 328
30.1. Negative Multinomial Distributions, 328
30.2. Stopping, 330
Exercises, 332
31. Sampling 334
31.1. Completely Random Sampling, 335
31.2. Counts, 335
31.3. Multiple Counts, 336
31.4. Statistical Sampling, 338
31.5. Stratified Sampling, 341
31.6. Random Attributes, 344
31.7. Cluster Sampling, 346
31.7.1. Single Stage Cluster Sampling, 347
31.7.2. Two Stage Cluster Sampling, 347
31.8. Ratio Estimates, 349
31.9. Linear Regression Estimators, 351
Exercises, 352
32. Markov Chains 358
32.1. Introduction, 358
32.2. Examples, 361
32.3. Strong Markov Property, 363
32.4. Transient and Recurrent States, 366
32.5. Decomposition of the State Space, 328
32.6. Hitting Distributions, 376
32.7. Transient and Recurrent Chains, 377
32.7.1. Random Walks, 379
32.8. Numbers of Visits to a State, 380
32.9. Invariant Measures, 382
32.10. Convergence to the Stationary Distributions, 390
32.11. Potential Theory, 395
32.11.1. Excessive Functions, 396
32.11.2. Potential Principles, 399
32.11.3. Transient and Recurrent Sets, 400
32.12. Reversible Chains, 404
Exercises, 407
CONTENTS Xiii
33. Discrete Renewal Processes 416
33.1. Definitions and Examples, 416
33.2. Discrete Renewal Theorem, 418
33.3. Renewal Sequences, 421
33.4. Modified Renewal Processes, 422
33.5. Some Fluctuation Theory, 425
33.5.1. Equivalence Principle, 426
33.5.2. Ladder Points, 429
Exercises, 436
PART III NONDISCRETE MODELS 441
34. Some Univariate Distributions 443
34.1. Uniform Distribution, 443
34.2. Gamma Distributions, 444
34.4. Normal Distributions, 446
34.4. Exponential Models, 449
Exercises, 450
35. Change of Variables 454
Exercises, 463
36. Beta Distributions 466
36.1. Definition and Basic Properties, 466
36.2. Connection with Bernoulli Trials, 469
Exercises, 470
37. Sampling Distributions 473
37.1. Three Basic Sampling Distributions, 473
37.2. Transformations of the Normal Distribution, 474
Exercises, 477
38. Dirichlet Distributions 481
38.1. The Dirichlet Distributions, 481
38.2. Ordered Dirichlet Distributions, 484
Exercises, 486
Xiv CONTENTS
39. Order Statistics 488
39.1. Basic Properties, 488
39.2. Homogeneous Poisson Processes on (0, oo), 494
39.3. Exponential Distributions, 496
39.4. Ranks, 497
39.5. Falling Below X*, 499
39.6. Record Values, 502
39.7. Coverages, 510
Exercises, 511
PART IV MULTIVARIATE NORMAL MODELS 519
40. Introduction 621
41. Representations 524
42. Singular and Nonsingular Distributions 526
43. Independence 530
Exercises, 532
44. Conditioning 534
Exercises, 541
45. Distributions of |X|2 542
Exercises, 546
46. Matrix Theory 548
PART V LIMIT CONCEPTS 553
47. Convergence with Probability One 555
47.1. Basic Properties, 555
47.2. Some Strong Convergence Results, 564
47.3. Applications to Renewal Theory, 570
47.4. Some Extensions of the Borel Cantelli Lemma, 573
47.5. Upcrossing and Submartingale Convergence, 576
47.6. Reverse Submartingales, 581
Exercises, 583
CONTENTS XV
48. Applications to Statistics 591
48.1. Empirical Distribution, 591
48.2. Sample Quantities, 593
Exercises, 594
49. Convergence in Probability 595
49.1. Properties of Convergence in Probability, 595
49.2. Weak Laws of Large Numbers, 599
49.3. Applications to Exchangeable Indicators, 602
49.4. Fair Games and Weak Laws of Large Numbers, 604
Exercises, 608
50. Convergence of Distributions 613
50.1. Definitions, 613
50.2. Properties of Convergence in Distribution, 614
50.3. Determining Convergence in Distribution, 622
50.4. Helly s Theorem, 625
50.5. Convergence of Types, 627
50.6. Triangular Arrays of Indicators, 628
50.7. Nonnegative Infinitely Divisible Distributions, 631
50.7.1. Limits of Triangular Arrays, 631
50.7.2. Infinitely Divisible Distributions, 635
50.7.3. Stable Distributions, 639
Exercises, 642
51. Characteristic Functions 648
51.1. Basic Properties, 648
51.2. Expansions of Characteristic Functions, 658
51.3. Characterizations of Characteristic Functions, 663
51.4. Inversion Formulas, 668
51.5. Weak Law of Large Numbers, 671
51.6. An Application to Recurrent Events, 672
Exercises, 675
52. Central Limit Theorem 68
52.1. The Classical Central Limit Theorem, 681
52.2. Some Applications of the CLT, 683
52.3. Lindeberg Theorem, 692
xvi CONTENTS
52.4. Random Sums, 703
52.5. Feller s Theorem, 704
52.6. Local Limit Theorems, 706
Exercises, 711
53. Limit Laws for Quantities 716
Exercises, 718
54. Limit Laws for Extreme Order Statistics 720
Exercises, 726
55. The Lp Spaces 728
55.1. Uniform Integrability, 728
55.2. Convergence in Lp, 731
55.3. Holder and Minkowski Inequalities, 734
55.4. Lp as a Normed Space, 736
55.5. Geometry of L2, 737
55.6. Existence of Conditional Expectation, 741
55.7. Uniform Integrability and Martingales, 742
Exercises, 746
56. Moment Generating Functions 750
56.1. Properties of a Moment Generating Function, 750
56.2. Applications to Random Walks, 755
Exercises, 757
57. Zero One Laws 758
57.1. Tail and Symmetric Events, 758
57.2. Kolmogorov s Zero One Law, 759
57.3. Exchangeable Sequences, 760
57.4. De Finetti s Theorem, 764
Exercises, 766
58. Random Series of Random Variables 768
58.1. Independent Random Variables, 768
58.2. Sums of Dependent Random Variables, 774
Exercises, 775
CONTENTS xvii
59. Stationary Sequences and the Ergodic Theorem 779
59.1. Introduction, 779
59.2. Measure Preserving Transformations, 779
59.3. Invariant Random Variables, 782
59.4. The Ergodic Theorem, 785
59.5. Range of a Random Walk, 789
59.6. Recurrence Properties, 791
59.7. Subadditive Ergodic Theorem, 792
Exercises, 800
60. Large Deviations Principle 803
Exercises, 810
61. Central Limit Theorems for Dependent Random Variables 811
61.1. The CLT for Sums Involving Random Permutations, 811
61.2. Applications to Sampling Theory, 817
61.3. A Martingale CLT, 820
61.4. Central Limit Theorems for Markov Chains, 823
Exercises, 825
62. Absolutely Continuous and Singular Probability Measures 826
62.1. Definitions, 826
62.2. Radon Nikodym Theorem, 827
62.3. Lebesgue Decomposition, 829
62.4. Equivalence for Distributions of Sequences of Independent
Random Vectors, 832
Exercises, 834
APPENDICES 837
1. Facts on Sequences 839
2. / and n Systems 843
3. Outer Measures 845
4. Construction of Measures 848
5. Monotone Functions on R 851
xviii contents
6. Dyadic Expansions 855
7. Mappings of [0,1] onto Rd 857
8. The Cantor Set and the Cantor Function 859
9. Translates of Integrable Functions 861
10. Nonmeasurable Sets 863
11. Convex Functions on R 865
12. Some Integrals 868
13. Some Abelian Formulas 870
14. Some Functional Equations 876
15. Kronecker s Lemma 878
16. A Result from Number Theory 880
Table of Tail Probability Under Standard Normal Distribution 881
Bibliography 882
Index 883
|
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| id | DE-604.BV009112390 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:31:12Z |
| institution | BVB |
| isbn | 0471632163 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006041403 |
| oclc_num | 27726655 |
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| record_format | marc |
| series2 | Wiley series in probability and mathematical statistics A Wiley interscience publication |
| spelling | Port, Sidney C. Verfasser (DE-588)172320852 aut Theoretical probability for applications Sidney C. Port New York [u.a.] Wiley 1994 XVIII, 894 S. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and mathematical statistics A Wiley interscience publication Offering comprehensive coverage of modern probability theory (exclusive of continuous time stochastic processes), this unique book functions as both an introduction for graduate statisticians, mathematicians, engineers, and economists and an encyclopedic reference of the subject for professionals in these fields. It assumes only a knowledge of calculus as well as basic real analysis and linear algebra Throughout Theoretical Probability for Applications the focus is on the practical uses of this increasingly important tool. It develops topics of discrete time probability theory for use in a multitude of applications, including stochastic processes, theoretical statistics, and other disciplines that require a sound foundation in modern probability theory. Principles of measure theory related to the study of probability theory are developed as they are required throughout the book The book examines most of the basic probability models that involve only a finite or countably infinite number of random variables. Topics in the "Discrete Models" section include Bernoulli trials, random walks, matching, sums of indicators, multinomial trials. Poisson approximations and processes, sampling. Markov chains, and discrete renewal theory. Nondiscrete models discussed include univariate, Beta, sampling, and Dirichlet distributions as well as order statistics Probabilités ram Waarschijnlijkheidstheorie gtt Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006041403&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Port, Sidney C. Theoretical probability for applications Probabilités ram Waarschijnlijkheidstheorie gtt Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4079013-7 |
| title | Theoretical probability for applications |
| title_auth | Theoretical probability for applications |
| title_exact_search | Theoretical probability for applications |
| title_full | Theoretical probability for applications Sidney C. Port |
| title_fullStr | Theoretical probability for applications Sidney C. Port |
| title_full_unstemmed | Theoretical probability for applications Sidney C. Port |
| title_short | Theoretical probability for applications |
| title_sort | theoretical probability for applications |
| topic | Probabilités ram Waarschijnlijkheidstheorie gtt Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Probabilités Waarschijnlijkheidstheorie Probabilities Wahrscheinlichkeitstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006041403&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT portsidneyc theoreticalprobabilityforapplications |