Probability: an introduction
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2014
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | X, 270 S. graph. Darst. |
| ISBN: | 9780198709978 9780198709961 |
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Record in the Search Index
| _version_ | 1804152457453895680 |
|---|---|
| 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
Divariate
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
В
Further Probability
6
Multlvariate 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
Divariate
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
С
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
χ
Contents
Appendix
A
Elements
of combinatorics
250
Appendix
В
Difference equations
252
Answers to exercises
255
Remarks on problems
259
Reading list
266
Index
267
Probability is an area of mathematics of tremendous contemporary
importance across all aspects of human endeavour. This book is a
compact account of the basic features of probability and random processes
at the level of first and second year mathematics undergraduates and Masters
students in cognate fields. It is suitable for a first course in probability, plus a follow-up
course in random processes including Markov chains.
A special feature is the authors attention to rigorous mathematics: not everything is rigorous,
but the need for rigour is explained at difficult junctures. The text is enriched by simple exercises,
together with problems (with very brief hints) many of which are taken from examinations at
Cambridge and Oxford.
The first eight chapters form a course in basic probability, being an account of events, random
variables, and distributions—discrete and continuous random variables are treated separately
—
together with simple versions of the law of large numbers and the central limit theorem. There is
an account of moment generating functions and their applications. The following three chapters
are about branching processes, random walks, and continuous-time random processes such as the
Poisson
process. The final chapter is a fairly extensive account of Markov chains in discrete time.
This second edition develops the success of the first edition through an updated presentation,
the extensive new chapter on Markov chains, and a number of new sections to ensure
comprehensive coverage of the syllabi at major universities.
Geoffrey Grimmett is Professor of Mathematical Statistics, University of Cambridge,
and Master of Downing College, Cambridge
Dominic Welsh is Professor Emeritus of Mathematics, University of Oxford,
and Emeritus Fellow of Merton College, Oxford
ALSO PUBLISHED BY OXFORD UNIVERSITY PRESS
Probability and Random Processes, Third Edition
Geoffrey Grimmett and David Stirzaker
One Thousand Exercises in Probability, Second Edition
Geoffrey Grimmett and David Stirzaker
|
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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 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| spelling | Grimmett, Geoffrey 1950- Verfasser (DE-588)120872919 aut Probability an introduction Geoffrey Grimmett ; Dominic Welsh 2. ed. Oxford [u.a.] Oxford Univ. Press 2014 X, 270 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 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 Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027476029&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027476029&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 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 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)4079013-7 (DE-588)4151278-9 |
| 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 |
| topic | Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Probabilities Wahrscheinlichkeitsrechnung Wahrscheinlichkeitstheorie Einführung |
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