Probability: an introduction
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford University Press
2017
|
| Edition: | Second edition, corrected reprint |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 270 Seiten Diagramme |
| ISBN: | 9780198709978 9780198709961 |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Part A Basic Probability
1 Events and probabilities 3
1.1 Experiments with chance.................................................. 3
1.2 Outcomes and events...................................................... 3
1.3 Probabilities............................................................ 6
1.4 Probability spaces....................................................... 7
1.5 Discrete sample spaces . . .............................................. 9
1.6 Conditional probabilities............................................... 11
1.7 Independent events...................................................... 12
1.8 The partition theorem................................................... 14
1.9 Probability measures are continuous..................................... 16
1.10 Worked problems ........................................................ 17
1.11 Problems ............................................................... 19
2 Discrete random variables 23
2.1 Probability mass functions.............................................. 23
2.2 Examples................................................................ 26
2.3 Functions of discrete random variables.................................. 29
2.4 Expectation........................................................... 30
2.5 Conditional expectation and the partition theorem.............. . 33
2.6 Problems............................................................... 35
3 Multivariate discrete distributions and independence 38
3.1 Bivariate discrete distributions........................................ 38
3.2 Expectation in the multivariate case................................... 40
3.3 Independence of discrete random variables............................... 41
3.4 Sums of random variables....................................... 44
3.5 Indicator functions..................................................... 45
3.6 Problems............................................................... 47
4 Probability generating functions 50
4.1 Generating functions.................................................... 50
4.2 Integer-valued random variables......................................... 51
4.3 Moments................................................................. 54
4.4 Sums of independent random variables.................................... 56
4.5 Problems............................................................... 58
viii Contents
5 Distribution functions and density functions 61
5.1 Distribution functions................................................... 61
5.2 Examples of distribution functions ...................................... 64
5.3 Continuous random variables.............................................. 65
5.4 Some common density functions............................................ 68
5.5 Functions of random variables............................................ 71
5.6 Expectations of continuous random variables.............................. 73
5.7 Geometrical probability.................................................. 76
5.8 Problems................................................................ 79
Part B Further Probability
6 Multivariate distributions and independence 83
6.1 Random vectors and independence.................... ............ 83
6.2 Joint density functions....................................... . 85
6.3 Marginal density functions and independence............................. 88
6.4 Sums of continuous random variables..................................... 91
6.5 Changes of variables.................................................... 93
6.6 Conditional density functions............................................ 95
6.7 Expectations of continuous random variables.............................. 97
6.8 Bivariate normal distribution..................... . . ......... 100
6.9 Problems............................................................... 102
7 Moments, and moment generating functions 108
7.1 A general note............................... .................. 108
7.2 Moments................................................................ Ill
7.3 Variance and covariance................................................. 113
7.4 Moment generating functions..................................... 117
7.5 Two inequalities........................................................ 121
7.6 Characteristic functions............................................... 125
7.7 Problems............................................................. 129
8 The main limit theorems 134
8.1 The law of averages..................................................... 134
8.2 Chebyshev’s inequality and the weak law . . ............................ 136
8.3 The central limit theorem............................................... 139
8.4 Large deviations and Cramer’s theorem.................................. 142
8.5 Convergence in distribution, and characteristic functions............... 145
8.6 Problems........................................................ . 149
Contents ix
Part C Random Processes
9 Branching processes 157
9.1 Random processes...................................................... 157
9.2 A model for population growth......................................... 158
9.3 The generating-function method........................................ 159
9.4 An example .................................................... 161
9.5 The probability of extinction......................................... 163
9.6 Problems.............................................................. 165
10 Random walks 167
10.1 One-dimensional random walks.......................................... 167
10.2 Transition probabilities ............................................. 168
10.3 Recurrence and transience of random walks............................. 170
10.4 The Gambler’s Ruin Problem............................................ 173
10.5 Problems.............................................................. 177
11 Random processes in continuous time 181
11.1 Life at a telephone switchboard....................................... 181
11.2 Poisson processes..................................................... 183
11.3 Inter-arrival times and the exponential distribution.................. 187
11.4 Population growth, and the simple birth process....................... 189
11.5 Birth and death processes............................................. 193
11.6 A simple queueing model............................................... 195
11.7 Problems.............................................................. 200
12 Markov chains 205
12.1 The Markov property .................................................. 205
12.2 Transition probabilities ............................................. 208
12.3 Class structure....................................................... 212
12.4 Recurrence and transience............................................. 214
12.5 Random walks in one, two, and three dimensions........................ 217
12.6 Hitting times and hitting probabilities............................... 221
12.7 Stopping times and the strong Markov property......................... 224
12.8 Classification of states.............................................. 227
12.9 Invariant distributions............................................... 231
12.10 Convergence to equilibrium............................................ 235
12.11 Time reversal......................................................... 240
12.12 Random walk on a graph................................................ 244
12.13 Problems.............................................................. 246
x Contents
Appendix A Elements of combinatorics 250
Appendix B Difference equations 252
Answers to exercises 255
Remarks on problems 259
Reading list 266
Index
267
|
| any_adam_object | 1 |
| author | Grimmett, Geoffrey 1950- Welsh, Dominic James Anthony |
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| author_role | aut aut |
| author_sort | Grimmett, Geoffrey 1950- |
| author_variant | g g gg d j a w dja djaw |
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| bvnumber | BV044655671 |
| classification_rvk | SK 800 |
| ctrlnum | (OCoLC)1013495231 (DE-599)BVBBV044655671 |
| 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 | Mathematik |
| edition | Second edition, corrected reprint |
| format | Book |
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| spelling | Grimmett, Geoffrey 1950- Verfasser (DE-588)120872919 aut Probability an introduction Geoffrey Grimmett ; Dominic Welsh Second edition, corrected reprint Oxford Oxford University Press 2017 X, 270 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Welsh, Dominic James Anthony Verfasser (DE-588)1089150164 aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030053298&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Grimmett, Geoffrey 1950- Welsh, Dominic James Anthony Probability an introduction Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
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| title | Probability an introduction |
| title_auth | Probability an introduction |
| title_exact_search | Probability an introduction |
| title_full | Probability an introduction Geoffrey Grimmett ; Dominic Welsh |
| title_fullStr | Probability an introduction Geoffrey Grimmett ; Dominic Welsh |
| title_full_unstemmed | Probability an introduction Geoffrey Grimmett ; Dominic Welsh |
| title_short | Probability |
| title_sort | probability an introduction |
| title_sub | an introduction |
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