Introduction to probability theory and stochastic processes:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, NJ
Wiley
2013
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index Weitere Ausgabe: Online version : Introduction to probability theory and stochastic processes |
| Beschreibung: | XXII, 959 S. graph. Darst. |
| ISBN: | 9781118382790 9781118615102 |
Internformat
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| 100 | 1 | |a Chiasson, John Nelson |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to probability theory and stochastic processes |c John Chiasson |
| 264 | 1 | |a Hoboken, NJ |b Wiley |c 2013 | |
| 300 | |a XXII, 959 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 500 | |a Includes bibliographical references and index | ||
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| 650 | 4 | |a Probabilities | |
| 650 | 4 | |a Stochastic processes | |
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Datensatz im Suchindex
| _version_ | 1804150472798371840 |
|---|---|
| adam_text | Titel: Introduction to probability theory and stochastic processes
Autor: Chiasson, John Nelson
Jahr: 2013
Contents
1 Coin Tossing 1
1.1 Coin Tossing Probability Model................ 1
1.2 Combinatorics: ...................... 3
1.2.1 Multinomial Formula.................. 6
1.2.2 Occupancy Problems*1 ................ 8
1.3 Random Variables on ün ................... 10
1.4 Binomial Probability Distribution............... 12
1.5 Weak Law of Large Numbers for Coin Tossing........ 17
1.6 Consistency Between Probability Spaces........... 18
1.7 Random Walks: Reflection Principle*............. 20
1.7.1 The Reflection Principle................ 22
1.8 Random Walks: First Returns and the Main Theorem* ... 25
1.8.1 First Returns...................... 29
1.8.2 Main Theorem..................... 32
1.9 Path Maximums and First Visits* .............. 36
1.10 Random Walks: Last Visits and Long Leads*........ 38
1.10.1 Inverse Sine Approximation for ct2k,2n........ 40
1.10.2 Long Leads....................... 43
1.10.3 Experimental Illustration............... 47
Appendix: Venn Diagrams...................... 50
Problems............................... 51
2 Countable Sample Spaces 61
2.1 Countable and Uncountable Sets............... 61
2.2 Probability for Countable Sample Spaces........... 65
2.2.1 Basic Properties of a Probability Space ....... 70
2.3 Examples: Finite Sample Spaces ............... 71
2.3.1 Biased Coins and the Binomial Distribution..... 76
2.3.2 Multinomial Probability Distribution ........ 79
2.4 Examples: Countably Infinite Sample Spaces ........ 81
2.4.1 Poisson Probability Distribution........... 87
Problems............................... 95
3 Conditional Probability in Countable Sample Spaces 105
1 Sections marked with an asterisk may be skipped without loss of continuity.
3.1 Conditional Probability.................... 105
3.2 Examples: Conditional Probability.............. 110
3.3 Independence.......................... 123
3.4 Coin Tossing and the SLLN.................. 129
Problems............................... 133
4 Uncountable Sample Spaces 151
4.1 Integration and Probability Theory.............. 153
4.1.1 The Borel Subsets ß{01]................ 154
4.1.2 Properties of /3(0 ^................... 155
4.1.3 TheProbability Space ((0,1],/3(01],P)........ 155
4.2 Modeling Coin Tossing on ((0,1], ^(0)1],P).......... 158
4.2.1 Dyadic Intervals....... ............. 162
4.2.2 Binomial Distribution................. 165
4.2.3 Independence of the Coin Tossing Random Variables 166
4.3 Independent Uniform RVs on ((0,1],/3(0)1],P)........ 168
4.3.1 Uo2 and Ue2........... ........... 168
4.3.2 Uo3 and Ue3....................... 175
4.3.3 General Procedure for Uon and Uen.......... 180
4.4 Formal Definition of Probability Space............ 183
4.4.1 Basic Properties of a Probability Space....... 188
Appendix: The Lebesgue Integral.................. 192
Appendix: U0 and Ue Are Independent and Uniform....... 198
Appendix: The SLLN and Uncountable Sets............ 201
Problems............................... 203
5 Continuous Random Variables 213
5.1 Induced Probability Distributions............... 213
5.2 Independent Random Variables................ 216
5.2.1 Independent Normal Random Variables....... 218
5.2.2 Independent Bernoulli Random Variables...... 222
5.2.3 Independent Poisson Random Variables....... 224
5.2.4 RVs with a Specified Probability Distribution* ... 225
5.3 Cumulative Distribution Functions.............. 227
5.4 Example: Random Arrival Times............... 232
Appendix - Proof of the Properties of CDFs............ 235
Problems............................... 237
6 Expectation 245
6.1 Expectation of Continuous Random Variables........ 245
6.2 Expectation of the Product of Independent RVs ...... 260
Problems............................... 264
7 Modeling Random Phenomena 267
7.1 Infinite Sequence of Independent Random Variables .... 267
7.2 Random Variables and Physical Phenomena ........ 270
7.2.1 One-Dimensional Navigation System......... 270
7.2.2 Waiting Times..................... 274
7.2.3 Bernoulli Random Process and Arrival Times . . . 278
7.2.4 Binary Communication System............ 285
7.2.5 Statistics........................ 287
7.2.6 Random Walks and Gambler s Ruin*......... 289
7.3 The SLLN and Relative Frequency.............. 300
Problems............................... 304
8 Functions of One Random Variable and Transforms 321
8.1 Functions of One Random Variable.............. 321
8.2 Inequalities........................... 330
8.3 Transforms........................... 335
8.4 Random Walks and Generating Functions* ......... 347
Problems............................... 354
9 Functions of Two Random Variables 365
9.1 Joint Distributions....................... 365
9.2 Expectation........................... 373
9.3 Independence.......................... 377
9.4 One Function of Two Random Variables........... 381
9.4.1 Sum of Two Random Variables............ 381
9.4.2 More Examples..................... 392
9.5 Poisson Process......................... 403
9.6 Central Limit Theorem .................... 407
Appendix - Leibniz Rule for Differentiation............ 420
Problems............................... 421
10 Two Functions of Two Random Variables 431
10.1 Transformation of Two Functions of Two Random Variables 431
10.1.1 General Affine Transformation............ 439
10.1.2 Affine Transformation of Jointly Normal RVs .... 441
10.2 Covariance and Correlation.................. 447
10.3 Moments and Transforms of Joint Distributions....... 455
Appendix: Change of Variables for Mulitple Integrals....... 460
Problems............................... 463
11 Conditional Probability for Continuous Random Variables473
11.1 Conditional Probability Density Functions.......... 473
11.2 Application: Binary Transmitter ............... 483
11.3 More Examples of Conditioning................ 488
11.4 Mean Square Estimation.................... 496
11.4.1 The Conditional Mean as the Best MS Estimate . . 498
11.5 Application: Navigation System with GPS ......... 502
11.6 Orthogonality Principle and MS Estimation ........ 509
11.6.1 Orthogonality Property of Conditional Expectation . 510
11.6.2 Linear MS Estimation and Orthogonality...... 513
11.7 Random Sums*......................... 524
11.7.1 Probability Density Function of Random Sums . . . 525
11.7.2 Direct Computation of the Mean and Variance . . . 526
11.7.3 Mean and Variance of Random Sums......... 528
11.7.4 Moment Generating Functions of Random Sums . . 530
Problems ............................... 532
12 Random Vectors 549
12.1 Joint Densities, Conditional Joint Densities, and Covariances 549
12.2 Definition of a Normal Random Vector............ 559
12.2.1 Linear Transformations of Normal Random Vectors . 561
12.2.2 Derivation of the Normal Density Function..... 562
12.2.3 Uncorrelated Normal Random Vectors........ 567
12.3 Conditional Density Functions of Jointly Normal RVs . . . 569
12.4 Linear Mean Square Estimation................ 574
Problems ............................... 581
13 Bernoulli, Geometrie, and Poisson Processes 587
13.1 Bernoulli and Geometrie Random Processes......... 587
13.1.1 Nonstationary Interarrival Waiting Time*...... 597
13.1.2 Fresh Start Property of Geometrie RVs*....... 601
13.2 Discrete Time Poisson Process ................ 604
13.3 The Poisson Process...................... 607
13.4 Order Statistics ........................ 618
13.5 Shot Noise ........................... 623
Problems............................... 629
14 Brownian Motion and White Noise 645
14.1 Random Walks in R...................... 645
14.2 Brownian Motion and the Wiener Process.......... 648
14.2.1 Fokker-Planck Equation ............... 653
14.2.2 Wiener Process from Random Sums*......... 655
14.3 Gaussian White Noise..................... 658
14.4 Thermal Gaussian White Noise................ 662
14.5 Discrete-Time Gaussian White Noise............. 668
14.5.1 Discrete Time to Continuous Time.......... 672
Appendix: The Dirac Delta Function................ 677
Problems............................... 700
15 Stationary Random Processes 703
15.1 Stationarity........................... 703
15.2 Overview of LTI Discrete-Time Systems........... 708
15.3 Stationarity and Discrete-Time Linear Systems....... 719
15.4 PSD of Discrete-Time Stationary Processes......... 724
15.5 Overview of LTI Continuous-Time Systems......... 737
15.6 Stationarity and Continuous-Time Linear Systems..... 744
15.7 PSD of Continuous-Time Stationary Processes ....... 747
15.8 Thermal White Gaussian Noise................ 757
15.9 Complex-Valued Stationary Processes............ 763
Appendix - Low-Pass Filters with Linear Phase.......... 765
Problems............................... 770
16 Convergence of Random Variables 777
16.1 Weak and Strong Law of Large Numbers........... 777
16.2 Central Limit Theorem .................... 783
16.3 Types of Convergence..................... 785
16.4 Borel-Cantelli Lemma..................... 805
16.5 Construction of Brownian Motion............... 808
16.5.1 Complete Orthonormal Functions........... 808
16.5.2 Construction of Brownian Motion on [0,1]...... 818
16.5.3 Construction of Brownian Motion on [0,oo]..... 826
16.5.4 Uniform Convergence of the Levy-Ciesielski Con-
struction ........................ 827
Problems............................... 832
17 Statistics 839
17.1 Gamma Distribution...................... 842
17.2 Variance of a Normal Random Variable ........... 846
17.3 Mean of a Normal Random Variable............. 853
17.4 Maximum Likelihood Estimation............... 859
17.5 Sufficient Statistics....................... 874
17.6 Goodness-of-Fit Test for Distributions............ 890
17.6.1 Multinomial Distribution............... 892
17.6.2 Probability Distribution of ^Li (i^Üf) ..... 894
Problems............................... 903
18 Kaiman Filter 905
18.1 Model.............................. 905
18.2 Innovations Approach to the Kaiman Filter......... 909
18.2.1 Innovations Process .................. 910
18.2.2 Kaiman Filter via the Innovations Process...... 912
18.3 Examples............................ 917
18.4 Optimal Linear Mean Square Estimator........... 921
18.5 The Extended Kaiman Filter................. 923
18.6 Aided Inertial Navigation Using GPS............. 924
Problems............................... 930
Further Reading 933
Table of Common Distributions 935
References 941
Index 946
|
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| isbn | 9781118382790 9781118615102 |
| language | English |
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| spelling | Chiasson, John Nelson Verfasser aut Introduction to probability theory and stochastic processes John Chiasson Hoboken, NJ Wiley 2013 XXII, 959 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Weitere Ausgabe: Online version : Introduction to probability theory and stochastic processes Probabilities Stochastic processes Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026071642&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chiasson, John Nelson Introduction to probability theory and stochastic processes Probabilities Stochastic processes Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4079013-7 |
| title | Introduction to probability theory and stochastic processes |
| title_auth | Introduction to probability theory and stochastic processes |
| title_exact_search | Introduction to probability theory and stochastic processes |
| title_full | Introduction to probability theory and stochastic processes John Chiasson |
| title_fullStr | Introduction to probability theory and stochastic processes John Chiasson |
| title_full_unstemmed | Introduction to probability theory and stochastic processes John Chiasson |
| title_short | Introduction to probability theory and stochastic processes |
| title_sort | introduction to probability theory and stochastic processes |
| topic | Probabilities Stochastic processes Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Probabilities Stochastic processes Wahrscheinlichkeitstheorie |
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