Probability theory, random processes and mathematical statistics:
The second part (Chapters 4-6) provides a foundation of stochastic analysis, gives information on basic models of random processes and tools to study them. Here a certain familiarity with elements of functional analysis is necessary. Important material is presented in the form of examples to keep re...
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch Russisch |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer
1995
|
| Schriftenreihe: | Mathematics and its applications
344 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007330197&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Zusammenfassung: | The second part (Chapters 4-6) provides a foundation of stochastic analysis, gives information on basic models of random processes and tools to study them. Here a certain familiarity with elements of functional analysis is necessary. Important material is presented in the form of examples to keep readers involved. Audience: This is a concise textbook for a graduate level course, with carefully selected topics representing the most important areas of modern probability, random processes and statistics |
| Abstract: | The study of random phenomena encountered in the real world is based on probability theory, mathematical statistics and the theory of random processes. The choice of the most suitable mathematical model is made on the basis of statistical data collected by observations. These models provide numerous tools for the analysis, prediction, and, ultimately, control of random phenomena. The first part of the present volume (Chapters 1-3) can serve as a self-contained, elementary introduction to probability, random processes and statistics. It contains a number of relatively simple and typical examples of random phenomena which allow a natural introduction of general structures and basic knowledge of elements of real/complex analysis, linear algebra and ordinary differential equations is required here |
| Beschreibung: | Aus dem Russ. übers. |
| Umfang: | VII, 255 S. Illustrationen |
| ISBN: | 0792337646 9780792337645 |
Internformat
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| 245 | 1 | 0 | |a Probability theory, random processes and mathematical statistics |c by Yu. A. Rozanov |
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| 490 | 1 | |a Mathematics and its applications |v 344 | |
| 500 | |a Aus dem Russ. übers. | ||
| 520 | 3 | |a The study of random phenomena encountered in the real world is based on probability theory, mathematical statistics and the theory of random processes. The choice of the most suitable mathematical model is made on the basis of statistical data collected by observations. These models provide numerous tools for the analysis, prediction, and, ultimately, control of random phenomena. The first part of the present volume (Chapters 1-3) can serve as a self-contained, elementary introduction to probability, random processes and statistics. It contains a number of relatively simple and typical examples of random phenomena which allow a natural introduction of general structures and basic knowledge of elements of real/complex analysis, linear algebra and ordinary differential equations is required here | |
| 520 | |a The second part (Chapters 4-6) provides a foundation of stochastic analysis, gives information on basic models of random processes and tools to study them. Here a certain familiarity with elements of functional analysis is necessary. Important material is presented in the form of examples to keep readers involved. Audience: This is a concise textbook for a graduate level course, with carefully selected topics representing the most important areas of modern probability, random processes and statistics | ||
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0001 96 A 2248 |
|---|---|
| DE-BY-TUM_katkey | 769500 |
| DE-BY-TUM_location | Mag |
| DE-BY-TUM_media_number | 040001996172 |
| _version_ | 1821938369323597824 |
| adam_text | CONTENTS
Preface ix
Annotation xi
Chapter 1. Introductory Probability Theory 1
1. The Notion of Probability 1
1.1. Equiprobable outcomes 1
1.2. Examples 2
1.3. Conditional probability 4
1.4. Independent events 7
1.5. Probability and frequency 9
2. Some Probability Models 10
2.1. Trials with countable outcomes 10
2.2. Bernoulli trials 11
2.3. Limit Poisson distribution 12
2.4. Finite number of events 15
2.5. The general model of probability theory 17
2.6. Some examples 26
3. Random Variables 32
3.1. Probability distributions 32
3.2. Joint probability distribution 35
3.3. Independent random variables 38
3.4. Conditional distributions 41
3.5. Functions of random variables 42
3.6. Random variables in the general model of probability theory 44
4. Mathematical Expectation 45
4.1. Mean value of discrete variable 45
4.2. Limit mean values 51
4.3. Some limit properties 54
4.4. Conditional expectation 60
5. Correlation 62
5.1. Variance and correlation 62
5.2. Normal correlations 66
5.3. Properties of the variance and the law of large numbers 69
6. Characteristic Functions 73
6.1. Some examples 73
6.2. Elementary analysis of characteristic functions 78
vi
6.3. The inverse formula of probability distributions 80
6.4. Weak convergence of distributions 82
7. The Central Limit Theorem 83
7.1. Some limit properties of probabilities 83
7.2. The central limit theorem 87
Chapter 2. Random Processes 91
1. Random Processes with Discrete State Space 91
1.1. The Poisson process and related processes 91
1.2. The Kolmogorov equations 96
1.3. Example (Branching processes) 100
1.4. The (limit) stationary probability distribution 107
2. Random Processes with Continuous States 113
2.1. The Brownian motion 113
2.2. Trajectories of the Brownian motion 115
2.3. Maxima and hitting times 122
2.4. Diffusion processes 126
Chapter 3. An Introduction to Mathematical Statistics 131
1. Some Examples of Statistical Problems and Methods 131
1.1. Estimation of the success probability in Bernoulli trials 131
1.2. Estimation of parameters in a normal sample 133
1.3. Chi square criterion for probability testing 137
1.4. Sequential analysis of alternative hypotheses 141
1.5. Bayesian approach to hypotheses testing and parameters estimation 144
1.6. Maximum likelihood method 147
1.7. Sample distribution function and the method of moments 149
1.8. The method of least squares 151
2. Optimality of Statistical Decisions 154
2.1. The most powerful criterion 154
2.2. Sufficient statistics 156
2.3. Lower bound for the mean square error 163
2.4. Asymptotic normality and efficiency of the maximum likelihood
estimate 166
Chapter 4. Basic Elements of Probability Theory 171
1. General Probability Distributions 171
1.1. Mappings and cr algebras 171
1.2. Approximation of events 175
1.3. 0 1 law 178
1.4. Mathematical expectation as the Lebesgue integral 179
1.5. £p spaces 181
2. Conditional Probabilities and Expectations 187
CONTENTS vii
2.1. Preliminary remarks 187
2.2. Conditional expectation and its properties 189
2.3. Conditional probability 191
3. Conditional Expectations and Martingales 194
3.1. General properties 194
Chapter 5. Elements of Stochastic Analysis and Stochastic Differential
Equations 201
1. Stochastic Series 201
1.1. Series of independent random variables 201
1.2. Three series criterion 203
2. Stochastic Integrals 207
2.1. Random functions (Preliminary remarks) 207
2.2. Integration in Cx space 209
2.3. Stochastic integrals in £2 space 212
2.4. Stochastic Ito integral in £2 space 218
3. Stochastic Integral Representations 222
3.1. Canonical representations 222
3.2. Spectral representation of a stationary process
and its applications 227
3.3. Stochastic integral representation of a process
with independent increments 231
4. Stochastic Differential Equations 237
4.1. Stochastic differentials 237
4.2. Linear stochastic differential equations 238
4.3. Linear differential equations with constant coefficients 242
4.4. The Kalman Bucy filter 246
Subject Index 253
|
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| author | Rozanov, Jurij A. 1934- |
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| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
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| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010958280 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:03:50Z |
| institution | BVB |
| isbn | 0792337646 9780792337645 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007330197 |
| oclc_num | 33104228 |
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| owner_facet | DE-739 DE-91 DE-BY-TUM DE-188 DE-83 |
| physical | VII, 255 S. Illustrationen |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spellingShingle | Rozanov, Jurij A. 1934- Probability theory, random processes and mathematical statistics Mathematics and its applications Analyse stochastique Jussieu Probabilités ram Processus stochastiques ram Statistique mathématique ram Mathematical statistics Probabilities Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Statistik (DE-588)4056995-0 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4057630-9 (DE-588)4064324-4 (DE-588)4056995-0 (DE-588)4079013-7 (DE-588)4151278-9 |
| title | Probability theory, random processes and mathematical statistics |
| title_alt | Lekcii po teorii verojatnostej |
| title_auth | Probability theory, random processes and mathematical statistics |
| title_exact_search | Probability theory, random processes and mathematical statistics |
| title_full | Probability theory, random processes and mathematical statistics by Yu. A. Rozanov |
| title_fullStr | Probability theory, random processes and mathematical statistics by Yu. A. Rozanov |
| title_full_unstemmed | Probability theory, random processes and mathematical statistics by Yu. A. Rozanov |
| title_short | Probability theory, random processes and mathematical statistics |
| title_sort | probability theory random processes and mathematical statistics |
| topic | Analyse stochastique Jussieu Probabilités ram Processus stochastiques ram Statistique mathématique ram Mathematical statistics Probabilities Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Statistik (DE-588)4056995-0 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Analyse stochastique Probabilités Processus stochastiques Statistique mathématique Mathematical statistics Probabilities Stochastic processes Stochastischer Prozess Wahrscheinlichkeitsrechnung Statistik Wahrscheinlichkeitstheorie Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007330197&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT rozanovjurija lekciipoteoriiverojatnostej AT rozanovjurija probabilitytheoryrandomprocessesandmathematicalstatistics |
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