From elementary probability to stochastic differential equations with MAPLE:
The authors provide a fast introduction to probabilistic and statistical concepts necessary to understand the basic ideas and methods of stochastic differential equations. The book is based on measure theory which is introduced as smoothly as possible. It is intended for advanced undergraduate stude...
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2002
|
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | The authors provide a fast introduction to probabilistic and statistical concepts necessary to understand the basic ideas and methods of stochastic differential equations. The book is based on measure theory which is introduced as smoothly as possible. It is intended for advanced undergraduate students or graduates, not necessarily in mathematics, providing an overview and intuitive background for more advanced studies as well as some practical skills in the use of MAPLE in the context of probability and its applications. Although this book contains definitions and theorems, it differs from conventional mathematics books in its use of MAPLE worksheets instead of formal proofs to enable the reader to gain an intuitive understanding of the ideas under consideration. As prerequisites the authors assume a familiarity with basic calculus and linear algebra, as well as with elementary ordinary differential equations and, in the final chapter, simple numerical methods for such ODEs. Although statistics is not systematically treated, they introduce statistical concepts such as sampling, estimators, hypothesis testing, confidence intervals, significance levels and p-values and use them in a large number of examples, problems and simulations. |
| Beschreibung: | XVI, 310 S. Illustrationen |
| ISBN: | 3540426663 |
Internformat
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| 245 | 1 | 0 | |a From elementary probability to stochastic differential equations with MAPLE |c Sasha Cyganowski ; Peter Kloeden ; Jerzy Ombach |
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| 300 | |a XVI, 310 S. |b Illustrationen | ||
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| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| 520 | 3 | |a The authors provide a fast introduction to probabilistic and statistical concepts necessary to understand the basic ideas and methods of stochastic differential equations. The book is based on measure theory which is introduced as smoothly as possible. It is intended for advanced undergraduate students or graduates, not necessarily in mathematics, providing an overview and intuitive background for more advanced studies as well as some practical skills in the use of MAPLE in the context of probability and its applications. Although this book contains definitions and theorems, it differs from conventional mathematics books in its use of MAPLE worksheets instead of formal proofs to enable the reader to gain an intuitive understanding of the ideas under consideration. As prerequisites the authors assume a familiarity with basic calculus and linear algebra, as well as with elementary ordinary differential equations and, in the final chapter, simple numerical methods for such ODEs. Although statistics is not systematically treated, they introduce statistical concepts such as sampling, estimators, hypothesis testing, confidence intervals, significance levels and p-values and use them in a large number of examples, problems and simulations. | |
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| 650 | 4 | |a Wahrscheinlichkeitsrechnung - Maple <Programm> | |
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Datensatz im Suchindex
| _version_ | 1804129056875085824 |
|---|---|
| adam_text | SASHA CYGANOWSKI PETER KLOEDEN JERZY OMBACH FROM ELEMENTARY PROBABILITY
TO STOCHASTIC DIFFERENTIAL EQUATIONS WITH MAPLE SPRINGER TABLE OF
CONTENTS 1 PROBABILITY BASICS 1 1.1 THE DEFINITION OF PROBABILITY 1
1.1.1 TOSSING TWO DICE 1 1.1.2 A STANDARD APPROACH 4 1.1.3 PROBABILITY
SPACE 4 1.2 THE CLASSICAL SCHEME AND ITS EXTENSIONS 6 1.2.1 DRAWING WITH
AND WITHOUT REPLACEMENT 8 1.2.2 APPLICATIONS OF THE CLASSICAL SCHEME 9
1.2.3 INFINITE SEQUENCE OF EVENTS 12 1.3 GEOMETRIC PROBABILITY 14 1.3.1
AWKWARD CORNERS - BERTRAND S PARADOX 15 1.4 CONDITIONAL AND TOTAL
PROBABILITY 16 1.4.1 THE LAW OF TOTAL PROBABILITY 17 1.4.2 BAYES
THEOREM 20 1.5 INDEPENDENT EVENTS 22 1.5.1 CARTESIAN PRODUCT 24 1.5.2
THE BERNOULLI SCHEME 25 1.6 MAPLE SESSION 27 1.7 EXERCISES 29 2 MEASURE
AND INTEGRAL 33 2.1 MEASURE 33 2.1.1 FORMAL DEFINITIONS 34 2.1.2
GEOMETRIC PROBABILITY REVISITED 36 2.1.3 PROPERTIES OF MEASURE 37 2.2
INTEGRAL 38 2.2.1 THE RIEMANN INTEGRAL 38 2.2.2 THE STIELTJES INTEGRAL
41 2.2.3 MEASURABLE FUNCTIONS 42 2.2.4 THE LEBESGUE INTEGRAL 44 2.3
PROPERTIES OF THE INTEGRAL 47 2.3.1 INTEGRALS WITH RESPECT TO LEBESGUE
MEASURE 49 2.3.2 RIEMANN INTEGRAL VERSUS LEBESGUE INTEGRAL 49 2.3.3
INTEGRALS VERSUS DERIVATIVES 50 2.4 DETERMINING INTEGRALS 51 XIV TABLE
OF CONTENTS 2.5 MAPLE SESSION 52 2.6 EXERCISES 59 3 RANDOM VARIABLES AND
DISTRIBUTIONS 61 3.1 PROBABILITY DISTRIBUTIONS 63 3.1.1 DISCRETE
DISTRIBUTIONS 63 3.1.2 CONTINUOUS DISTRIBUTIONS 65 3.1.3 DISTRIBUTION
FUNCTION 66 3.2 RANDOM VARIABLES AND RANDOM VECTORS 69 3.2.1 A PROBLEM
OF MEASURABILITY 70 3.2.2 DISTRIBUTION OF A RANDOM VARIABLE 71 3.2.3
RANDOM VECTORS 72 3.3 INDEPENDENCE 72 3.4 FUNCTIONS OF RANDOM VARIABLES
AND VECTORS 74 3.4.1 DISTRIBUTIONS OF A SUM 77 3.5 MAPLE SESSION 78 3.6
EXERCISES 82 4 PARAMETERS OF PROBABILITY DISTRIBUTIONS 85 4.1
MATHEMATICAL EXPECTATION 85 4.2 VARIANCE 89 4.2.1 MOMENTS 90 4.3
COMPUTATION OF MOMENTS 92 4.3.1 EVALUATION OF THE MEAN AND VARIANCE 93
4.4 CHEBYSHEV S INEQUALITY 98 4.5 LAW OF LARGE NUMBERS 100 4.5.1 THE
WEAK LAW OF LARGE NUMBERS 101 4.5.2 CONVERGENCE OF RANDOM VARIABLES 102
4.5.3 THE STRONG LAW OF LARGE NUMBERS 106 4.6 CORRELATION 107 4.6.1
MARGINAL DISTRIBUTIONS 108 4.6.2 COVARIANCE AND THE COEFFICIENT OF
CORRELATION 108 4.7 MAPLE SESSION 112 4.8 EXERCISES 117 5 A TOUR OF
IMPORTANT DISTRIBUTIONS 121 5.1 COUNTING 121 5.1.1 THE BINOMIAL
DISTRIBUTION 121 5.1.2 THE MULTINOMIAL DISTRIBUTION 123 5.1.3 THE
POISSON DISTRIBUTION 124 5.1.4 THE HYPERGEOMETRIC DISTRIBUTION 127 5.2
WAITING TIMES 128 5.2.1 THE GEOMETRIC DISTRIBUTION 129 5.2.2 THE
NEGATIVE BINOMIAL DISTRIBUTION 130 5.2.3 THE EXPONENTIAL DISTRIBUTION
131 TABLE OF CONTENTS XV 5.2.4 THE ERLANG DISTRIBUTION 134 5.2.5 THE
POISSON PROCESS 135 5.3 THE NORMAL DISTRIBUTION 136 5.4 CENTRAL LIMIT
THEOREM 140 5.4.1 EXAMPLES 145 5.5 MULTIDIMENSIONAL NORMAL DISTRIBUTION
147 5.5.1 2-DIMENSIONAL NORMAL DISTRIBUTION 148 5.6 MAPLE SESSION 153
5.7 EXERCISES 158 NUMERICAL SIMULATIONS AND STATISTICAL INFERENCE 161
6.1 PSEUDO-RANDOM NUMBER GENERATION 161 6.2 BASIC STATISTICAL TESTS 165
6.2.1 P-VALUE 167 6.3 THE RUNS TEST 168 6.4 GOODNESS OF FIT TESTS 174
6.5 INDEPENDENCE TEST 177 6.6 CONFIDENCE INTERVALS 179 6.7 INFERENCE FOR
NUMERICAL SIMULATIONS 182 6.8 MAPLE SESSION 185 6.9 EXERCISES 190
STOCHASTIC PROCESSES 193 7.1 CONDITIONAL EXPECTATION 193 7.2 MARKOV
CHAINS 198 7.2.1 RANDOM WALKS 199 7.2.2 EVOLUTION OF PROBABILITIES 202
7.2.3 IRREDUCIBLE AND TRANSIENT CHAINS 205 7.3 SPECIAL CLASSES OF
STOCHASTIC PROCESSES 210 7.4 CONTINUOUS-TIME STOCHASTIC PROCESSES 211
7.4.1 WIENER PROCESS 212 7.4.2 MARKOV PROCESSES 215 7.4.3 DIFFUSION
PROCESSES 217 7.5 CONTINUITY AND CONVERGENCE 218 7.6 MAPLE SESSION 220
7.7 EXERCISES 226 STOCHASTIC CALCULUS 229 8.1 INTRODUCTION 229 8.2 ITO
STOCHASTIC INTEGRALS 230 8.3 STOCHASTIC DIFFERENTIAL EQUATIONS 233 8.4
STOCHASTIC CHAIN RULE: THE ITO FORMULA 236 8.5 STOCHASTIC TAYLOR
EXPANSIONS 238 8.6 STRATONOVICH STOCHASTIC CALCULUS 241 8.7 MAPLE
SESSION 244 XVI TABLE OF CONTENTS 8.8 EXERCISES 246 9 STOCHASTIC
DIFFERENTIAL EQUATIONS 249 9.1 SOLVING SCALAR STRATONOVICH SDES 249 9.2
LINEAR SCALAR SDES 253 9.2.1 MOMENT EQUATIONS 256 9.3 SCALAR SDES
REDUCIBLE TO LINEAR SDES 257 9.4 VECTOR SDES 260 9.4.1 VECTOR ITO
FORMULA 262 9.4.2 FOKKER-PLANCK EQUATION 264 9.5 VECTOR LINEAR SDE 265
9.5.1 MOMENT EQUATIONS 267 9.5.2 LINEARIZATION 267 9.6 VECTOR
STRATONOVICH SDES 268 9.7 MAPLE SESSION 270 9.8 EXERCISES 275 10
NUMERICAL METHODS FOR SDES 277 10.1 NUMERICAL METHODS FOR ODES 277 10.2
THE STOCHASTIC EULER SCHEME 280 10.2.1 STATISTICAL ESTIMATES AND
CONFIDENCE INTERVALS 283 10.3 HOW TO DERIVE HIGHER ORDER SCHEMES 284
10.3.1 MULTIPLE STOCHASTIC INTEGRALS 286 10.4 HIGHER ORDER STRONG
SCHEMES 288 10.5 HIGHER ORDER WEAK SCHEMES 290 10.6 THE EULER AND
MILSTEIN SCHEMES FOR VECTOR SDES 291 10.7 MAPLE SESSION 295 10.8
EXERCISES 301 BIBLIOGRAPHICAL NOTES 303 REFERENCES 305 INDEX 307
|
| any_adam_object | 1 |
| author | Cyganowski, Sasha Kloeden, Peter E. 1949- Ombach, Jerzy |
| author_GND | (DE-588)115479155 |
| author_facet | Cyganowski, Sasha Kloeden, Peter E. 1949- Ombach, Jerzy |
| author_role | aut aut aut |
| author_sort | Cyganowski, Sasha |
| author_variant | s c sc p e k pe pek j o jo |
| building | Verbundindex |
| bvnumber | BV014180375 |
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| callnumber-label | QA273 |
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| callnumber-subject | QA - Mathematics |
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| ctrlnum | (OCoLC)248014669 (DE-599)BVBBV014180375 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| format | Book |
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| id | DE-604.BV014180375 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:59:04Z |
| institution | BVB |
| isbn | 3540426663 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009718670 |
| oclc_num | 248014669 |
| open_access_boolean | |
| owner | DE-M347 DE-703 DE-355 DE-BY-UBR DE-824 DE-29T DE-91G DE-BY-TUM DE-384 DE-945 DE-706 DE-634 DE-83 DE-11 DE-188 DE-578 |
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| physical | XVI, 310 S. Illustrationen |
| publishDate | 2002 |
| publishDateSearch | 2002 |
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| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Cyganowski, Sasha Verfasser aut From elementary probability to stochastic differential equations with MAPLE Sasha Cyganowski ; Peter Kloeden ; Jerzy Ombach Berlin [u.a.] Springer 2002 XVI, 310 S. Illustrationen txt rdacontent n rdamedia nc rdacarrier Universitext The authors provide a fast introduction to probabilistic and statistical concepts necessary to understand the basic ideas and methods of stochastic differential equations. The book is based on measure theory which is introduced as smoothly as possible. It is intended for advanced undergraduate students or graduates, not necessarily in mathematics, providing an overview and intuitive background for more advanced studies as well as some practical skills in the use of MAPLE in the context of probability and its applications. Although this book contains definitions and theorems, it differs from conventional mathematics books in its use of MAPLE worksheets instead of formal proofs to enable the reader to gain an intuitive understanding of the ideas under consideration. As prerequisites the authors assume a familiarity with basic calculus and linear algebra, as well as with elementary ordinary differential equations and, in the final chapter, simple numerical methods for such ODEs. Although statistics is not systematically treated, they introduce statistical concepts such as sampling, estimators, hypothesis testing, confidence intervals, significance levels and p-values and use them in a large number of examples, problems and simulations. Maple (Computer file) Equazioni differenziali stocastiche sbt Probabilità sbt Stochastische Differentialgleichung - Maple <Programm> Wahrscheinlichkeitsrechnung - Maple <Programm> Probabilities Stochastic differential equations Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Maple Programm (DE-588)4209397-1 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Maple Programm (DE-588)4209397-1 s DE-604 Stochastische Differentialgleichung (DE-588)4057621-8 s Kloeden, Peter E. 1949- Verfasser (DE-588)115479155 aut Ombach, Jerzy Verfasser aut HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009718670&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cyganowski, Sasha Kloeden, Peter E. 1949- Ombach, Jerzy From elementary probability to stochastic differential equations with MAPLE Maple (Computer file) Equazioni differenziali stocastiche sbt Probabilità sbt Stochastische Differentialgleichung - Maple <Programm> Wahrscheinlichkeitsrechnung - Maple <Programm> Probabilities Stochastic differential equations Stochastische Differentialgleichung (DE-588)4057621-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Maple Programm (DE-588)4209397-1 gnd |
| subject_GND | (DE-588)4057621-8 (DE-588)4064324-4 (DE-588)4209397-1 |
| title | From elementary probability to stochastic differential equations with MAPLE |
| title_auth | From elementary probability to stochastic differential equations with MAPLE |
| title_exact_search | From elementary probability to stochastic differential equations with MAPLE |
| title_full | From elementary probability to stochastic differential equations with MAPLE Sasha Cyganowski ; Peter Kloeden ; Jerzy Ombach |
| title_fullStr | From elementary probability to stochastic differential equations with MAPLE Sasha Cyganowski ; Peter Kloeden ; Jerzy Ombach |
| title_full_unstemmed | From elementary probability to stochastic differential equations with MAPLE Sasha Cyganowski ; Peter Kloeden ; Jerzy Ombach |
| title_short | From elementary probability to stochastic differential equations with MAPLE |
| title_sort | from elementary probability to stochastic differential equations with maple |
| topic | Maple (Computer file) Equazioni differenziali stocastiche sbt Probabilità sbt Stochastische Differentialgleichung - Maple <Programm> Wahrscheinlichkeitsrechnung - Maple <Programm> Probabilities Stochastic differential equations Stochastische Differentialgleichung (DE-588)4057621-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Maple Programm (DE-588)4209397-1 gnd |
| topic_facet | Maple (Computer file) Equazioni differenziali stocastiche Probabilità Stochastische Differentialgleichung - Maple <Programm> Wahrscheinlichkeitsrechnung - Maple <Programm> Probabilities Stochastic differential equations Stochastische Differentialgleichung Wahrscheinlichkeitsrechnung Maple Programm |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009718670&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT cyganowskisasha fromelementaryprobabilitytostochasticdifferentialequationswithmaple AT kloedenpetere fromelementaryprobabilitytostochasticdifferentialequationswithmaple AT ombachjerzy fromelementaryprobabilitytostochasticdifferentialequationswithmaple |