Fundamentals of applied probability and random processes:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam [u.a.]
Elsevier
2005
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 442 S. graph. Darst. |
| ISBN: | 0120885085 9780120885084 |
Internformat
MARC
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| 245 | 1 | 0 | |a Fundamentals of applied probability and random processes |c Oliver C. Ibe |
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2005 | |
| 300 | |a XVII, 442 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Lehrbuch | |
| 650 | 4 | |a Probabilities | |
| 650 | 4 | |a Stochastic processes | |
| 650 | 4 | |a Stochastischer Prozess | |
| 650 | 4 | |a Wahrscheinlichkeitsrechnung | |
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Datensatz im Suchindex
| _version_ | 1805084107607113728 |
|---|---|
| adam_text |
Contents
Preface xiii
Acknowledgment xvii
1 Basic Probability Concepts 1
1.1 Introduction 1
1.2 Sample Space and Events 2
1.3 Definitions of Probability 4
1.3.1 Axiomatic Definition 4
1.3.2 Relative Frequency Definition 5
1.3.3 Classical Definition 5
1.4 Applications of Probability 7
1.4.1 Reliability Engineering 7
1.4.2 Quality Control 7
1.4.3 Channel Noise 7
1.4.4 System Simulation 8
1.5 Elementary Set Theory 8
1.5.1 Set Operations 9
1.5.2 Number of Subsets of a Set 10
1.5.3 Venn Diagram 10
1.5.4 Set Identities 10
1.5.5 Duality Principle 13
1.6 Properties of Probability 13
1.7 Conditional Probability 15
1.7.1 Total Probability and the Bayes'Theorem 16
1.7.2 Tree Diagram 24
V
vi Contents
1.8 Independent Events 26
1.9 Combined Experiments 29
1.10 Basic Combinatorial Analysis 31
1.10.1 Permutations 31
1.10.2 Circular Arrangement 33
1.10.3 Applications of Permutations in Probability 33
1.10.4 Combinations 35
1.10.5 The Binomial Theorem 37
1.10.6 Stirling's Formula 37
1.10.7 Applications of Combinations in Probability 38
1.11 Reliability Applications 42
1.12 Chapter Summary 47
1.13 Problems 47
1.14 References 57
2 Random Variables 59
2.1 Introduction 59
2.2 Definition of a Random Variable 59
2.3 Events Defined by Random Variables 61
2.4 Distribution Functions 62
2.5 Discrete Random Variables 63
2.5.1 Obtaining the PMF from the CDF 68
2.6 Continuous Random Variables 70
2.7 Chapter Summary 75
2.8 Problems 76
3 Moments of Random Variables 85
3.1 Introduction 85
3.2 Expectation 86
3.3 Expectation of Nonnegative Random Variables 88
3.4 Moments of Random Variables and the Variance 90
3.5 Conditional Expectations 101
3.6 The Chebyshev Inequality 102
3.7 The Markov Inequality 103
3.8 Chapter Summary 104
3.9 Problems 104
Contents vii
4 Special Probability Distributions 111
4.1 Introduction 111
4.2 The Bernoulli Trial and Bernoulli Distribution 112
4.3 Binomial Distribution 113
4.4 Geometric Distribution 116
4.4.1 Modified Geometric Distribution 119
4.4.2 "Forgetfulness" Property of the Geometric Distribution 120
4.5 Pascal (or Negative Binomial) Distribution 122
4.6 Hypergeometric Distribution 126
4.7 Poisson Distribution 130
4.7.1 Poisson Approximation to the Binomial Distribution 132
4.8 Exponential Distribution 133
4.8.1 "Forgetfulness" Property of the Exponential Distribution 134
4.8.2 Relationship between the Exponential and Poisson
Distributions 136
4.9 Erlang Distribution 136
4.10 Uniform Distribution 141
4.10.1 The Discrete Uniform Distribution 142
4.11 Normal Distribution 144
4.11.1 Normal Approximation to the Binomial Distribution 147
4.11.2 The Error Function 149
4.11.3 The (2 Function 150
4.12 The Hazard Function 150
4.13 Chapter Summary 153
4.14 Problems 155
5 Multiple Random Variables 167
5.1 Introduction 167
5.2 Joint CDFs of Bivariate Random Variables 167
5.2.1 Properties of the Joint CDF 168
5.3 Discrete Random Variables 169
5.4 Continuous Random Variables 173
5.5 Determining Probabilities from a Joint CDF 175
5.6 Conditional Distributions 178
5.6.1 Conditional PMF for Discrete Random Variables 178
5.6.2 Conditional PDF for Continuous Random Variables 179
5.6.3 Conditional Means and Variances 180
5.6.4 Simple Rule for Independence 182
ii Contents
5.7 Covariance and Correlation Coefficient 184
5.8 Many Random Variables 187
5.9 Multinomial Distributions 189
5.10 Chapter Summary 190
5.11 Problems 190
6 Functions of Random Variables 197
6.1 Introduction 197
6.2 Functions of One Random Variable 198
6.2.1 Linear Functions 198
6.2.2 Power Functions 199
6.3 Expectation of a Function of One Random Variable 201
6.3.1 Moments of a Linear Function 201
6.4 Sums of Independent Random Variables 202
6.4.1 Moments of the Sum of Random Variables 209
6.4.2 Sum of Discrete Random Variables 210
6.4.3 Sum of Independent Binomial Random Variables 214
6.4.4 Sum of Independent Poisson Random Variables 214
6.4.5 The Spare Parts Problem 215
6.5 Minimum of Two Independent Random Variables 218
6.6 Maximum of Two Independent Random Variables 219
6.7 Comparison of the Interconnection Models 221
6.8 Two Functions of Two Random Variables 222
6.8.1 Application of the Transformation Method 224
6.9 Laws of Large Numbers 226
6.10 The Central Limit Theorem 227
6.11 Order Statistics 229
6.12 Chapter Summary 233
6.13 Problems 234
7 Transform Methods 241
7.1 Introduction 241
7.2 The Characteristic Function 242
7.2.1 Moment Generating Property of the Characteristic
Function 243
7.3 The s Transform 245
7.3.1 Moment Generating Property of the s Transform 245
7.3.2 The s Transforms of Some Weil Known PDFs 246
Contents IX
7.3.3 The s Transform of the PDF of the Sum of Independent
Random Variables 247
7.4 The z Transform 250
7.4.1 Moment Generating Property of the z Transform 252
7.4.2 The z Transform of the Bernoulli Distribution 253
7.4.3 The z Transform of the Binomial Distribution 254
7.4.4 The z Transform of the Geometric Distribution 254
7.4.5 The z Transform of the Poisson Distribution 255
7.4.6 The z Transform of the PMF of the Sum of Independent
Random Variables 255
7.4.7 The z Transform of the Pascal Distribution 256
7.5 Random Sum of Random Variables 256
7.6 Chapter Summary 261
7.7 Problems 261
8 Introduction to Random Processes 267
8.1 Introduction 267
8.2 Classification of Random Processes 269
8.3 Characterizing a Random Process 269
8.3.1 Mean and Autocorrelation Function of a Random Process 270
8.3.2 The Autocovariance Function of a Random Process 271
8.4 Crosscorrelation and Crosscovariance Functions 272
8.4.1 Review of Some Trigonometric Identities 273
8.5 Stationary Random Processes 275
8.5.1 Strict Sense Stationary Processes 275
8.5.2 Wide Sense Stationary Processes 275
8.6 Ergodic Random Processes 282
8.7 Power Spectral Density 284
8.7.1 White Noise 289
8.8 Discrete Time Random Processes 290
8.8.1 Mean, Autocorrelation Function, and Autocovariance
Function 290
8.8.2 Power Spectral Density 291
8.8.3 Sampling of Continuous Time Processes 292
8.9 Chapter Summary 293
8.10 Problems 294
X Contents
9 Linear Systems with Random Inputs 305
9.1 Introduction 305
9.2 Overview of Linear Systems with Deterministic Inputs 305
9.3 Linear Systems with Continuous Time Random Inputs 307
9.4 Linear Systems with Discrete Time Random Inputs 313
9.5 Autoregressive Moving Average Process 316
9.5.1 Moving Average Process 316
9.5.2 Autoregressive Process 319
9.5.3 ARMA Process 322
9.6 Chapter Summary 323
9.7 Problems 323
10 Some Models of Random Processes 333
10.1 Introduction 333
10.2 The Bernoulli Process 333
10.3 Random Walk 335
10.3.1 Gambler's Ruin 337
10.4 The Gaussian Process 339
10.4.1 White Gaussian Noise Process 341
10.5 Poisson Process 342
10.5.1 Counting Processes 342
10.5.2 Independent Increment Processes 343
10.5.3 Stationary Increments 343
10.5.4 Definitions of a Poisson Process 344
10.5.5 Interarrival Times for the Poisson Process 345
10.5.6 Conditional and Joint PMFs for Poisson Processes 346
10.5.7 Compound Poisson Process 347
10.5.8 Combinations of Independent Poisson Processes 349
10.5.9 Competing Independent Poisson Processes 350
10.5.10 Subdivision of a Poisson Process and the Filtered Poisson
Process 352
10.5.11 Random Incidence 353
10.5.12 Nonhomogeneous Poisson Process 356
10.6 Markov Processes 358
10.7 Discrete Time Markov Chains 359
10.7.1 State Transition Probability Matrix 360
10.7.2 The « Step State Transition Probability 360
10.7.3 State Transition Diagrams 361
Contents xi
10.7.4 Classification of States 363
10.7.5 Limiting State Probabilities 366
10.7.6 Doubly Stochastic Matrix 369
10.8 Continuous Time Markov Chains 370
10.8.1 Birth and Death Processes 373
10.9 Gambler's Ruin as a Markov Chain 376
10.10 Chapter Summary 378
10.11 Problems 378
11 Introduction to Statistics 395
11.1 Introduction 395
11.2 Sampling Theory 396
11.2.1 The Sample Mean 397
11.2.2 The Sample Variance 399
11.2.3 Sampling Distributions 400
11.3 Estimation Theory 402
11.3.1 Point Estimate, Interval Estimate, and Confidence
Interval 403
11.3.2 Maximum Likelihood Estimation 405
11.3.3 Minimum Mean Squared Error Estimation 408
11.4 Hypothesis Testing 411
11.4.1 Hypothesis Test Procedure 412
11.4.2 Type I and Type II Errors 413
11.4.3 One Tailed and Two Tailed Tests 413
11.5 Curve Fitting and Linear Regression 418
11.6 Chapter Summary 422
11.7 Problems 422
Appendix 1: Table for the CDF of the Standard Normal Random
Variable 427
Bibliography 429
Index 433 |
| any_adam_object | 1 |
| author | Ibe, Oliver C. 1947- |
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| dewey-ones | 519 - Probabilities and applied mathematics |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| spelling | Ibe, Oliver C. 1947- Verfasser (DE-588)136641784 aut Fundamentals of applied probability and random processes Oliver C. Ibe Amsterdam [u.a.] Elsevier 2005 XVII, 442 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lehrbuch Probabilities Stochastic processes Stochastischer Prozess Wahrscheinlichkeitsrechnung Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Stochastischer Prozess (DE-588)4057630-9 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013836524&sequence=000002&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 | Ibe, Oliver C. 1947- Fundamentals of applied probability and random processes Lehrbuch Probabilities Stochastic processes Stochastischer Prozess Wahrscheinlichkeitsrechnung Stochastischer Prozess (DE-588)4057630-9 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
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| title | Fundamentals of applied probability and random processes |
| title_auth | Fundamentals of applied probability and random processes |
| title_exact_search | Fundamentals of applied probability and random processes |
| title_full | Fundamentals of applied probability and random processes Oliver C. Ibe |
| title_fullStr | Fundamentals of applied probability and random processes Oliver C. Ibe |
| title_full_unstemmed | Fundamentals of applied probability and random processes Oliver C. Ibe |
| title_short | Fundamentals of applied probability and random processes |
| title_sort | fundamentals of applied probability and random processes |
| topic | Lehrbuch Probabilities Stochastic processes Stochastischer Prozess Wahrscheinlichkeitsrechnung Stochastischer Prozess (DE-588)4057630-9 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Lehrbuch Probabilities Stochastic processes Stochastischer Prozess Wahrscheinlichkeitsrechnung |
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