Probability and conditional expectation: fundamentals for the empirical sciences
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Chichester, West Sussex
John Wiley & Sons, Inc.
2017
|
| Schriftenreihe: | Wiley series in probability and statistics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes index |
| Beschreibung: | xviii, 576 Seiten Illustrationen |
| ISBN: | 9781119243526 |
Internformat
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| 245 | 1 | 0 | |a Probability and conditional expectation |b fundamentals for the empirical sciences |c Rolf Steyer (Institute of Psychology, University of Jena, Germany), Werner Nagel (Institute of Mathematics, University of Jena, Germany) |
| 264 | 1 | |a Chichester, West Sussex |b John Wiley & Sons, Inc. |c 2017 | |
| 264 | 4 | |c © 2017 | |
| 300 | |a xviii, 576 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Wiley series in probability and statistics | |
| 500 | |a Includes index | ||
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| 650 | 4 | |a Measure theory | |
| 650 | 4 | |a Measure algebras | |
| 650 | 4 | |a Probabilities | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-029754498 | ||
Datensatz im Suchindex
| _version_ | 1804177593204736000 |
|---|---|
| adam_text | Contents
Preface xv
Acknowledgements xix
About the companion website xxi
Part I Measure-Theoretical Foundations of Probability Theory 1
1 Measure 3
1.1 Introductory examples 3
1.2 a-Algebra and measurable space 4
1.2.1 a-Algebra generated by a set system 9
1.2.2 O AIgebra of Borel sets on 12
1.2.3 a-Algebra on a Cartesian product 14
1.2.4 n-Stable set systems that generate a a-algebra 15
1.3 Measure and measure space 17
1.3.1 o Additivity and related properties 18
1.3.2 Other properties 19
1.4 Specific measures 20
1.4.1 Dirac measure and counting measure 21
1.4.2 Lebesgue measure 22
1.4.3 Other examples of a measure 23
1.4.4 Finite and a-finite measures 24
1.4.5 Product measure 24
1.5 Continuity of a measure 25
1.6 Specifying a measure via a generating system 27
1.7 o-Algebra that is trivial with respect to a measure 28
1.8 Proofs 28
Exercises 31
Solutions 32 2
2 Measurable mapping 42
2.1 Image and inverse image 42
2.2 Introductory examples 43
2.2.1 Example 1: Rectangles 43
2.2.2 Example 2: Flipping two coins 45
2.3 Measurable mapping 47
2.3.1 Measurable mapping 47
2.3.2 a-Algebra generated by a mapping 52
2.3.3 Final a-algebra 55
V1U
CONTENTS
2.3.4 Multivariate mapping 55
2.3.5 Projection mapping 57
2.3.6 Measurability with respect to a mapping 58
2.4 Theorems on measurable mappings 59
2.4.1 Measurability of a composition 60
2.4.2 Theorems on measurable functions 62
2.5 Equivalence of two mappings with respect to a measure 65
2.6 Image measure 69
2.7 Proofs 72
Exercises 78
Solutions 78
3 Integral 87
3.1 Definition 87
3.1.1 Integral of a nonnegative step function 87
3.1.2 Integral of a nonnegative measurable function 92
3.1.3 Integral of a measurable function 97
3.2 Properties 100
3.2.1 Integral of//-equivalent functions 102
3.2.2 Integral with respect to a weighted sum of measures 104
3.2.3 Integral with respect to an image measure 106
3.2.4 Convergence theorems 107
3.3 Lebesgue and Riemann integral 108
3.4 Density 110
3.5 Absolute continuity and the Radon-Nikodym theorem 112
3.6 Integral with respect to a product measure 114
3.7 Proofs 115
Exercises 124
Solutions 125
Part II Probability, Random Variable, and its Distribution 131
4 Probability measure 133
4.1 Probability measure and probability space 133
4.1.1 Definition 133
4.1.2 Formal and substantive meaning of probabilistic terms 134
4.1.3 Properties of a probability measure 134
4.1.4 Examples 135
4.2 Conditional probability 138
4.2.1 Definition 138
4.2.2 Filtration and time order between events and sets of events 139
4.2.3 Multiplication rule 141
4.2.4 Examples 142
4.2.5 Theorem of total probability 143
4.2.6 Bayes’ theorem 144
4.2.7 Conditional-probability measure 145
CONTENTS
IX
4.3 Independence 149
4.3.1 Independence of events 149
4.3.2 Independence of set systems 150
4.4 Conditional independence given an event 151
4.4.1 Conditional independence of events given an event 152
4.4.2 Conditional independence of set systems given an event 152
4.5 Proofs 154
Exercises 156
Solutions 157
5 Random variable, distribution, density, and distribution function 162
5.1 Random variable and its distribution 162
5.2 Equivalence of two random variables with respect to a probability measure 168
5.2.1 Identical and P-equivalent random variables 168
5.2.2 P-equivalence, P^-equivalence, and absolute continuity 171
5.3 Multivariate random variable 174
5.4 Independence of random variables 176
5.5 Probability function of a discrete random variable 182
5.6 Probability density with respect to a measure 185
5.6.1 General concepts and properties 186
5.6.2 Density of a discrete random variable 187
5.6.3 Density of a bivariate random variable 188
5.7 Uni- or multivariate real-valued random variable 189
5.7.1 Distribution function of a univariate real-valued random variable 189
5.7.2 Distribution function of a multivariate real-valued random variable 192
5.7.3 Density of a continuous univariate real-valued random variable 193
5.7.4 Density of a continuous multivariate real-valued random variable 195
5.8 Proofs 196
Exercises 204
Solutions 205
6 Expectation, variance, and other moments 208
6.1 Expectation 208
6.1.1 Definition 208
6.1.2 Expectation of a discrete random variable 209
6.1.3 Computing the expectation using a density 211
6.1.4 Transformation theorem 212
6.1.5 Rules of computation 216
6.2 Moments, variance, and standard deviation 216
6.3 Proofs 221
Exercises 221
Solutions 222
7 Linear quasi-regression, covariance, and correlation 225
7.1 Linear quasi-regression 225
7.2 Covariance 228
7.3 Correlation 232
X
CONTENTS
7.4 Expectation vector and covariance matrix 235
7.4.1 Random vector and random matrix 235
7.4.2 Expectation of a random vector and a random matrix 235
7.4.3 Covariance matrix of two multivariate random variables 237
7.5 Multiple linear quasi-regression 238
7.6 Proofs 240
Exercises 245
Solutions 245
8 Some distributions 254
8.1 Some distributions of discrete random variables 254
8.1.1 Discrete uniform distribution 254
8.1.2 Bernoulli distribution 255
8.1.3 Binomial distribution 256
8.1.4 Poisson distribution 258
8.1.5 Geometric distribution 260
8.2 Some distributions of continuous random variables 262
8.2.1 Continuous uniform distribution 262
8.2.2 Normal distribution 264
8.2.3 Multivariate normal distribution 267
8.2.4 Central -distribution 271
8.2.5 Central ¿-distribution 273
8.2.6 Central F-distribution 274
8.3 Proofs 276
Exercises 280
Solutions 281
Part III Conditional Expectation and Regression 285
9 Conditional expectation value and discrete conditional expectation 287
9.1 Conditional expectation value 287
9.2 Transformation theorem 290
9.3 Other properties 292
9.4 Discrete conditional expectation 294
9.5 Discrete regression 295
9.6 Examples 296
9.7 Proofs 301
Exercises 302
Solutions 302
10 Conditional expectation 305
10.1 Assumptions and definitions 305
10.2 Existence and uniqueness 307
10.2.1 Uniqueness with respect to a probability measure 308
10.2.2 A necessary and sufficient condition of uniqueness 309
10.2.3 Examples 310
CONTENTS
xi
10.3 Rules of computation and other properties 311
10.3.1 Rules of computation 311
10.3.2 Monotonicity 314
10.3.3 Convergence theorems 314
10.4 Factorization, regression, and conditional expectation value 316
10.4.1 Existence of a factorization 316
10.4.2 Conditional expectation and mean squared error 317
10.4.3 Uniqueness of a factorization 318
10.4.4 Conditional expectation value 319
10.5 Characterizing a conditional expectation by the joint distribution 322
10.6 Conditional mean independence 323
10.7 Proofs 328
Exercises 331
Solutions 331
11 Residual, conditional variance, and conditional covariance 340
11.1 Residual with respect to a conditional expectation 340
11.2 Coefficient of determination and multiple correlation 345
11.3 Conditional variance and covariance given a o-algebra 350
11.4 Conditional variance and covariance given a value of a random variable 351
11.5 Properties of conditional variances and covariances 354
11.6 Partial correlation 357
11.7 Proofs 359
Exercises 360
Solutions 361
12 Linear regression 369
12.1 Basic ideas 369
12.2 Assumptions and definitions 371
12.3 Examples 373
12.4 Linear quasi-regression 378
12.5 Uniqueness and identification of regression coefficients 380
12.6 Linear regression 381
12.7 Parameterizations of a discrete conditional expectation 383
12.8 Invariance of regression coefficients 387
12.9 Proofs 388
Exercises 390
Solutions 391
13 Linear logistic regression 393
13.1 Logit transformation of a conditional probability 393
13.2 Linear logistic parameterization 396
13.3 A parameterization of a discrete conditional probability 398
13.4 Identification of coefficients of a linear logistic parameterization 399
13.5 Linear logistic regression and linear logit regression 400
13.6 Proofs 407
Exercises 409
Solutions 409
xii CONTENTS
14 Conditional expectation with respect to a conditional-probability measure 412
14.1 Introductory examples 413
14.2 Assumptions and definitions 417
14.3 Properties 423
14.4 Partial conditional expectation 424
14.5 Factorization 426
14.5.1 Conditional expectation value with respect to PB 426
14.5.2 Uniqueness of factorizations 427
14.6 Uniqueness 428
14.6.1 A necessary and sufficient condition of uniqueness 428
14.6.2 Uniqueness with respect to P and other probability measures 430
14.6.3 Necessary and sufficient conditions of P-uniqueness 430
14.6.4 Properties related to P-uniqueness 433
14.7 Conditional mean independence with respect to Pz=z 437
14.8 Proofs 439
Exercises 444
Solutions 444
15 Effect functions of a discrete regressor 450
15.1 Assumptions and definitions 450
15.2 Intercept function and effect functions 451
15.3 Implications of independence of X and Z for regression coefficients 454
15.4 Adjusted effect functions 456
15.5 Logit effect functions 460
15.6 Implications of independence of X and Z for the logit regression
coefficients 463
15.7 Proofs 466
Exercises 468
Solutions 469
Part IV Conditional Independence and Conditional Distribution 471
16 Conditional independence 473
16.1 Assumptions and definitions 473
16.1.1 Two events 474
16.1.2 Two sets of events 475
16.1.3 Two random variables 476
16.2 Properties 477
16.3 Conditional independence and conditional mean independence 485
16.4 Families of events 487
16.5 Families of set systems 488
16.6 Families of random variables 489
16.7 Proofs 493
Exercises 501
Solutions 501
CONTENTS
xiii
17 Conditional distribution 505
17.1 Conditional distribution given a o-algebra or a random variable 505
17.2 Conditional distribution given a value of a random variable 508
17.3 Existence and uniqueness 511
17.3.1 Existence 511
17.3.2 Uniqueness of the functions Py ( A ) 512
17.3.3 Common null set uniqueness of a conditional distribution 513
17.4 Conditional-probability measure given a value of a random variable 516
17.5 Decomposing the joint distribution of random variables 518
17.6 Conditional independence and conditional distributions 520
17.7 Expectations with respect to a conditional distribution 525
17.8 Conditional distribution function and probability density 528
17.9 Conditional distribution and Radon-Nikodym density 531
17.10 Proofs 534
Exercises 553
Solutions 553
References 557
List of Symbols 559
Author Index 569
Subject Index 570
|
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| id | DE-604.BV044351726 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:50:32Z |
| institution | BVB |
| isbn | 9781119243526 |
| language | English |
| lccn | 016025874 |
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| spelling | Steyer, Rolf 1950- Verfasser (DE-588)130635324 aut Probability and conditional expectation fundamentals for the empirical sciences Rolf Steyer (Institute of Psychology, University of Jena, Germany), Werner Nagel (Institute of Mathematics, University of Jena, Germany) Chichester, West Sussex John Wiley & Sons, Inc. 2017 © 2017 xviii, 576 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Includes index Random variables Measure theory Measure algebras Probabilities Statistik (DE-588)4056995-0 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Statistik (DE-588)4056995-0 s DE-604 Nagel, Werner 1952- Verfasser (DE-588)1076883419 aut Erscheint auch als Online-Ausgabe (epub) 978-1-119-24348-9 Erscheint auch als Online-Ausgabe (epdf) 978-1-119-24350-2 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029754498&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Steyer, Rolf 1950- Nagel, Werner 1952- Probability and conditional expectation fundamentals for the empirical sciences Random variables Measure theory Measure algebras Probabilities Statistik (DE-588)4056995-0 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4056995-0 (DE-588)4079013-7 |
| title | Probability and conditional expectation fundamentals for the empirical sciences |
| title_auth | Probability and conditional expectation fundamentals for the empirical sciences |
| title_exact_search | Probability and conditional expectation fundamentals for the empirical sciences |
| title_full | Probability and conditional expectation fundamentals for the empirical sciences Rolf Steyer (Institute of Psychology, University of Jena, Germany), Werner Nagel (Institute of Mathematics, University of Jena, Germany) |
| title_fullStr | Probability and conditional expectation fundamentals for the empirical sciences Rolf Steyer (Institute of Psychology, University of Jena, Germany), Werner Nagel (Institute of Mathematics, University of Jena, Germany) |
| title_full_unstemmed | Probability and conditional expectation fundamentals for the empirical sciences Rolf Steyer (Institute of Psychology, University of Jena, Germany), Werner Nagel (Institute of Mathematics, University of Jena, Germany) |
| title_short | Probability and conditional expectation |
| title_sort | probability and conditional expectation fundamentals for the empirical sciences |
| title_sub | fundamentals for the empirical sciences |
| topic | Random variables Measure theory Measure algebras Probabilities Statistik (DE-588)4056995-0 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Random variables Measure theory Measure algebras Probabilities Statistik Wahrscheinlichkeitstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029754498&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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