Fundamentals of probability: a first course
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York ; Dordrecht ; Heidelberg ; London
Springer
[2010]
|
| Schriftenreihe: | Springer texts in statistics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 450 Seiten Diagramme |
| ISBN: | 9781441957795 9781441957801 |
Internformat
MARC
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| 245 | 1 | 0 | |a Fundamentals of probability |b a first course |c Anirban DasGupta |
| 264 | 1 | |a New York ; Dordrecht ; Heidelberg ; London |b Springer |c [2010] | |
| 264 | 4 | |c © 2010 | |
| 300 | |a XVI, 450 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer texts in statistics | |
| 650 | 0 | 7 | |a Wahrscheinlichkeitstheorie |0 (DE-588)4079013-7 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-020198952 | ||
Datensatz im Suchindex
| _version_ | 1804142788388847616 |
|---|---|
| adam_text | Contents
Preface
...............................................................................
vii
1
Introducing Probability
...................................................... 1
1.1
Experiments and Sample Spaces
..................................... 2
1.2
Set Theory Notation and Axioms of Probability
.................... 3
1.3
How to Interpret a Probability
........................................ 5
.,4
Calculating Probabilities
............................................. 7
1.4.1
Manual Counting
............................................ 8
1.4.2
General Counting Methods
................................. 10
1.5
Inclusion-Exclusion Formula
......................................... 12
1.6 *
Bounds on the Probability of a Union
............................. 15
1.7
Synopsis
............................................................... 16
1.8
Exercises
.............................................................. 16
References
...................................................................... 21
2
The Birthday and Matching Problems
..................................... 23
2.1
The Birthday Problem
................................................ 23
2.1.1 *
Stirling s Approximation
................................. 24
2.2
The Matching Problem
............................................... 25
2.3
Synopsis
............................................................... 26
2.4
Exercises
.............................................................. 27
References
...................................................................... 27
3
Conditional Probability and Independence
................................ 29
3.1
Basic Formulas and First Examples
.................................. 29
3.2
More Advanced Examples
........................................... 31
3.3
Independent Events
................................................... 33
3.4
Bayes Theorem
....................................................... 36
3.5
Synopsis
............................................................... 39
3.6
Exercises
.............................................................. 39
Contents
Integer-
Valued and Discrete Random Variables
.......................... 45
4.1
Mass Function
......................................................... 45
4.2
CDF and Median of a Random Variable
............................. 47
4.2.1
Functions of a Random Variable
........................... 53
4.2.2
Independence of Random Variables
........................ 55
4.3
Expected Value of a Discrete Random Variable
..................... 56
4.4
Basic Properties of Expectations
..................................... 57
4.5
Illustrative Examples
................................................. 59
4.6
Using Indicator Variables to Calculate Expectations
................ 60
4.7
The Tail Sum Method for Calculating Expectations
................ 62
4.8
Variance, Moments, and Basic Inequalities
.......................... 63
4.9
Illustrative Examples
................................................. 65
4.9.1
Variance of a Sum of Independent Random Variables
.... 67
4.10
Utility of
μ
and
σ
as Summaries
..................................... 67
4.10.1
Chebyshev s Inequality and the Weak Law
of Large Numbers
........................................... 68
4.10.2 *
Better Inequalities
......................................... 70
4.11 *
Other Fundamental Moment Inequalities
.......................... 71
4.11.1 *
Applying Moment Inequalities
........................... 73
4.12
Truncated Distributions
............................................... 74
4.13
Synopsis
............................................................... 75
4.14
Exercises
.............................................................. 76
References
...................................................................... 80
Generating Functions
........................................................ 81
5.1
Generating Functions
................................................. 81
5.2
Moment Generating Functions and
Cumulants
...................... 85
5.2.1 *
Cumulants.................................................
87
5.3
Synopsis
............................................................... 89
5.4
Exercises
.............................................................. 89
References
...................................................................... 90
Standard Discrete Distributions
............................................ 91
6.1
Introduction to Special Distributions
................................ 91
6.2
Discrete Uniform Distribution
....................................... 94
6.3
Binomial Distribution
................................................. 95
6.4
Geometric and Negative Binomial Distributions
.................... 99
6.5
Hypergeometric Distribution
.........................................102
6.6
Poisson
Distribution
..................................................104
6.6.
1 Mean Absolute Deviation and the Mode
...................108
6.7
Poisson
Approximation to Binomial
.................................109
6.8 *
Miscellaneous
Poisson
Approximations
...........................112
6.9
Benford sLaw
........................................................114
6.10
Distribution of Sums and Differences
................................115
6.10.1 *
Distribution of Differences
...............................117
Contents
6.11 *
Discrete
Does Not Mean Integer-Valued
..........................118
6.12
Synopsis
...............................................................119
6.13
Exercises
..............................................................121
References
......................................................................
1
25
Continuous Random Variables
..............................................127
7.1
The Density Function and the CDF
..................................127
7.1.1
Quantiles
.....................................................133
7.2
Generating New Distributions from Old
.............................135
7.3
Normal and Other Symmetric Unimodal Densities
.................137
7.4
Functions of a Continuous Random Variable
........................140
7.4.1
Quantile Transformation
....................................144
7.4.2
Cauchy Density
..............................................145
7.5
Expectation of Functions and Moments
.............................147
7.6
The Tail Probability Method for Calculating Expectations
.........155
7.6.1 *
Survival and Hazard Rate
.................................155
7.6.2 *
Moments and the Tail
.....................................155
7.7 *
Moment Generating Function and Fundamental Tail
Inequalities
............................................................157
7.7.1 *
Chernoff-Bernstein Inequality
............................158
7.7.2 *
Lugosi s Improved Inequality
............................160
7.8 *
Jensen and Other Moment Inequalities and a Paradox
............161
7.9
Synopsis
...............................................................163
7.10
Exercises
..............................................................165
References
......................................................................169
Some Special Continuous Distributions
....................................171
8.1
Uniform Distribution
.................................................171
8.2
Exponential and Weibuil Distributions
..............................173
8.3
Gamma and Inverse Gamma Distributions
..........................177
8.4
Beta Distribution
......................................................182
8.5
Extreme-Value Distributions
.........................................185
8.6 *
Exponential Density and the
Poisson
Process
.....................187
8.7
Synopsis
...............................................................190
8.8
Exercises
..............................................................191
References
......................................................................194
Normal Distribution
..........................................................195
9.1
Definition and Basic Properties
......................................195
9.2
Working with a Normal Table
........................................199
9.3
Additional Examples and the
Lognormal
Density
..................200
9.4
Sums of Independent Normal Variables
.............................203
9.5
Mills Ratio and Approximations for the Standard Normal CDF
.. .205
9.6
Synopsis
...............................................................208
9.7
Exercises
..............................................................209
References
......................................................................212
xiv Contents
10 Normal
Approximations
and the Central Limit Theorem
...............213
10.1
Some Motivating Examples
..........................................213
10.2
Central Limit Theorem
...............................................215
10.3
Normal Approximation to Binomial
.................................217
10.3.1
Continuity Correction
.......................................218
10.3.2
A New Rule of Thumb
......................................222
10.4
Examples of the General CLT
........................................224
10.5
Normal Approximation to
Poisson
and Gamma
.....................229
10.6 *
Convergence of Densities and Higher-Order Approximations
.. .232
10.6.1 *
Refined Approximations
..................................233
10.7
Practical Recommendations for Normal Approximations
..........236
10.8
Synopsis
...............................................................237
10.9
Exercises
..............................................................238
References
......................................................................242
11
Multivariate Discrete Distributions
.........................................243
11.1
Bivariate Joint Distributions and Expectations of Functions
.......243
11.2
Conditional Distributions and Conditional Expectations
...........250
11.2.1
Examples on Conditional Distributions
and Expectations
............................................251
11.3
Using Conditioning to Evaluate Mean and Variance
................255
11.4
Covariance and Correlation
..........................................258
11.5
Multivariate Case
.....................................................263
11.5.1 *
Joint MGF
.................................................264
11.5.2
Multinomial Distribution
...................................265
11.6
Synopsis
...............................................................268
11.7
Exercises
..............................................................270
12
Multidimensional Densities
..................................................275
12.1
Joint Density Function and Its Role
..................................275
12.2
Expectation of Functions
.............................................285
12.3
Bivariate Normal
......................................................289
12.4
Conditional Densities and Expectations
.............................294
12.4.
1 Examples on Conditional Densities and Expectations
... .296
12.5
Bivariate Normal Conditional Distributions
.........................302
12.6
Order Statistics
........................................................303
12.6.1
Basic Distribution Theory
..................................304
12.6.2 *
More Advanced Distribution Theory
....................306
12.7
Synopsis
...............................................................311
12.8
Exercises
..............................................................314
References
......................................................................319
Contents xv
13
Convolutions and Transformations
.........................................321
13.1
Convolutions and Examples
..........................................321
13.2
Products and Quotients and the
t
and
F
Distributions
..............326
13.3
Transformations
.......................................................330
13.4
Applications of the Jacobian Formula
...............................332
13.5
Polar Coordinates in Two Dimensions
...............................333
13.6
Synopsis
...............................................................336
13.7
Exercises
..............................................................337
References
......................................................................341
14
Markov Chains and Applications
...........................................343
14.1
Notation and Basic Definitions
.......................................344
14.2
Chapman-Kolmogorov Equation
.....................................349
14.3
Communicating Classes
..............................................353
14.4 *
Gambler s Ruin
.....................................................355
14.5 *
First Passage, Recurrence, and Transience
........................357
14.6
Long-Run Evolution and Stationary Distributions
..................363
14.7
Synopsis
...............................................................370
14.8
Exercises
..............................................................370
References
......................................................................378
15
Urn Models in Physics and Genetics
........................................379
15.1
Stirling Numbers and Their Basic Properties
........................379
15.2
Urn Models in Quantum Mechanics
.................................381
15.3 *
Poisson
Approximations
...........................................386
15.4
Pólya sUrn
............................................................388
15.5
Pólya-Eggenberger
Distribution
......................................390
15.6 *
de
Finetti s Theorem and
Pólya
Urns
..............................391
15.7
Urn Models in Genetics
..............................................393
15.7.1
Wright-Fisher Model
........................................393
15.7.2
Time until AHele Uniformity
...............................395
15.8
Mutation and Hoppe s Urn
...........................................396
15.9 *
The Ewens Sampling Formula
.....................................399
15.10
Synopsis
...............................................................401
15.11
Exercises
..............................................................403
References
......................................................................406
Appendix I: Supplementary Homework and Practice Problems
.............409
1.1 Word Problems
........................................................409
1.2
True-False Problems
..................................................426
xvi Contents
Appendix
II:
Symbols and
Formulas
............................................433
II.
1
Glossary of
Symbols..................................................433
11.2
Formula Summaries
..................................................436
11.
2.1
Moments and MGFs of Common Distributions
...........436
11.2.2 Useful Mathematical Formulas
.............................439
11.2.3 Useful Calculus Facts
.......................................440
11.3 Tables
..................................................................440
11.3.1 Normal Table
................................................440
11.3.2
Poisson
Table
................................................442
Author Index
........................................................................443
Subject Index
.......................................................................445
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| spelling | DasGupta, Anirban (DE-588)139121250 aut Fundamentals of probability a first course Anirban DasGupta New York ; Dordrecht ; Heidelberg ; London Springer [2010] © 2010 XVI, 450 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Springer texts in statistics Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Wahrscheinlichkeitstheorie (DE-588)4079013-7 s b DE-604 Erscheint auch als Online-Ausgabe 978-1-4419-5780-1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020198952&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | DasGupta, Anirban Fundamentals of probability a first course Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4123623-3 |
| title | Fundamentals of probability a first course |
| title_auth | Fundamentals of probability a first course |
| title_exact_search | Fundamentals of probability a first course |
| title_full | Fundamentals of probability a first course Anirban DasGupta |
| title_fullStr | Fundamentals of probability a first course Anirban DasGupta |
| title_full_unstemmed | Fundamentals of probability a first course Anirban DasGupta |
| title_short | Fundamentals of probability |
| title_sort | fundamentals of probability a first course |
| title_sub | a first course |
| topic | Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Wahrscheinlichkeitstheorie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020198952&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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