An introduction to probability theory with statistical applications:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Plenum Press
1984
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| Series: | Mathematical concepts and methods in science and engineering
29 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 662 S. Illustrationen |
| ISBN: | 030641645X |
Staff View
MARC
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| 245 | 1 | 0 | |a An introduction to probability theory with statistical applications |c Michael A. Golberg |
| 246 | 1 | 3 | |a Probability theory |
| 264 | 1 | |a New York, NY [u.a.] |b Plenum Press |c 1984 | |
| 300 | |a XI, 662 S. |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Mathematical concepts and methods in science and engineering |v 29 | |
| 650 | 4 | |a Estadística matemática | |
| 650 | 4 | |a Mathematical statistics | |
| 650 | 4 | |a Probabilities | |
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Record in the Search Index
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| adam_text | Contents
1. Experiments, Sets, and Sample Spaces
1.1. Introduction 1
1.2. Deterministic and Nondeterministic Experiments 2
1.3. Sets and Sample Spaces 3
1.4. Sets 5
1.4.1. Definitions and Basic Properties 5
1.4.2. Operations on Sets 8
1.4.3. Further Properties of Set Operations 12
1.5. Cartesian Products 18
1.6. Functions 23
1.7. Sequences and Series 28
1.7.1. Sequences 28
1.7.2. Limits 31
1.7.3. Infinite Series 35
Exercises 39
2. Probability Spaces
2.1. Introduction 45
2.2. Relative Frequencies and Finite Probability Spaces 46
2.3. Properties of Finite Probability Spaces 56
2.3.1. Probability of Events 56
2.3.2. Calculation Theorems 63
2.4. Denumerable Probability Spaces 75
2.5. Nondiscrete Probability Spaces and the Kolmogorov Axioms 81
Exercises 86
3. Counting and Uniform Probability Spaces
3.1. Introduction 93
3.2. Basic Counting Methods 94
3.2.1. Tree Diagrams 94
vii
viii Contents
3.2.2. The Fundamental Principle of Counting 97
3.3. Permutations and Combinations 105
3.3.1 Permutations 106
3.3.2. Combinations 115
3.4. Binomial Coefficients 121
3.5. Further Applications of Combinatorial Probability 126
3.5.1. Keno 126
3.5.2. Poker 129
3.5.3. The Poker Test: Is n Random? 132
3.5.4. The Match Problem 137
3.5.5. Multinomial Coefficients 140
Exercises 143
4. Conditional Probability
4.1. Introduction 149
4.2. Conditional Probability 149
4.3. The Product Rule 154
4.4. Total Probability and Conditioning 159
4.5. Bayes s Rule 163
Exercises 166
5. Independence
5.1. Introduction 169
5.2. Independent Events 169
5.3. Discrete Product Spaces 174
5.4. Bernoulli Trials 180
5.5. Introduction to Reliability Theory 185
5.6. Infinite Bernoulli Trials 191
Exercises 196
6. Discrete Random Variables
6.1. Introduction 201
6.2. Random Variables and Their Distributions 202
6.3. Some Commonly Occurring Distributions 213
6.3.1. Degenerate Distribution 213
6.3.2. Bernoulli Distribution 213
6.3.3. The Uniform Distribution 214
6.3.4. Binomial Distribution 214
6.3.5. Poisson Distribution 217
6.3.6. Geometric Distribution 224
6.3.7. Negative Binomial Distribution 226
6.3.8. Hypergeometric Distribution 229
6.3.9. Empirical Distributions and Zipfs Law 230
6.4. Computations Using the Distribution 233
Exercises 238
Contents ix
7. Joint Distributions and Independent Random Variables
7.1. Introduction 245
7.2. Joint Distribution 245
7.3. Independent Random Variables 262
7.4. Functions of Random Variables 269
7.4.1. Definition and Examples 269
7.4.2. Sums of Random Variables 275
7.4.3. Probability Generating Functions 277
7.4.4. Some Order Statistics 283
Exercises 285
8. Expectation
8.1. Introduction 291
8.2. Definition and Properties of Expectation 292
8.3. Expectation on Denumerable Sample Spaces 301
8.4. Expectation of Functions of Random Variables 306
8.5. Variance and Covariance 322
8.6. Chebyshev s Inequality and the Law of Large Numbers 337
8.7. Conditional Expectation 343
Exercises 346
9. Densities and The Central Limit Theorem
9.1. Introduction 353
9.2. Probability Spaces on the Real Line 354
9.3. Expectation 360
9.4. Normal Random Variables 365
9.5. The Central Limit Theorem 374
Exercises 381
10. Distribution of Nondiscrete Random Variables
10.1. Introduction 385
10.2. Cumulative Distribution Function 385
10.3. Some Commonly Occurring Densities 395
10.3.1. Gamma Function 396
10.3.2. Beta Function 397
10.3.3. Gamma Densities 399
10.3.4. Lognormal Densities 404
10.3.5. Weibull Densities 406
10.3.6. Beta Densities 409
10.3.7. T and F Densities 410
10.4. Functions of a Random Variable 411
Exercises 417
x Contents
11. Joint Distribution of Nondiscrete Random Variables
11.1. Introduction 423
11.2. Joint Densities 423
11.3. Independent Random Variables 437
11.4. Functions of Independent Random Variables 440
11.4.1. Distribution of Sums 440
11.4.2. Distribution of Ratios 445
11.4.3. Method of Jacobians 451
11.4.4. Some Normal Distribution Theory 456
11.5. Order Statistics 461
11.5.1. Distribution Theory 461
11.5.2. Reliability Theory 471
11.5.3. Spare Parts and the Renewal Equation 478
11.6. Generating Functions 484
11.6.1. Moment Generating Functions 485
11.6.2. Characteristic Functions 489
11.6.3. Weak Law of Large Numbers 495
11.6.4. Central Limit Theorem 497
11.7. Joint Generating Functions 501
11.7.1. Joint Probability Generating Functions 501
11.7.2. Joint Moment Generating Functions 502
11.7.3. The Joint Characteristic Function 505
11.8. Multivariate Central Limit Theorem 506
11.8.1. Multivariate Normal Densities 507
11.8.2. Proof of the Multivariate Central Limit Theorem 514
11.9. Conditional Densities 520
11.9.1. Introduction 520
11.9.2. Conditional Expectation 524
Exercises 526
12. Statistics
12.1. Introduction 537
12.2. Parameter Estimation 538
12.2.1. Estimation for Discrete Random Variables 539
12.2.2. Estimation for Absolutely Continuous Random Variables . . . 548
12.3. Properties of Estimators 559
12.3.1. Consistency and Unbiasedness 559
12.3.2. Sufficiency 563
12.4. Confidence Intervals 568
12.4.1. Exact Confidence Intervals 568
12.4.2. Approximate Confidence Intervals 573
12.5. Hypothesis Testing 575
12.5.1. Best Tests 577
12.5.2. Generalized Likelihood Ratio Tests 583
Contents xi
12.6. x2 Tests 587
12.6.1. Goodness of Fit 587
12.6.2. x2 Tests for Independence 592
12.7. Hypothesis Testing and Confidence Intervals 594
12.8. Linear Regression 595
12.8.1. The Normal Equations 596
12.8.2. Properties of $ 603
12.8.3. Testing: a2 Unknown 606
Exercises 612
Solutions to Selected Exercises 623
Appendix 647
References 651
Index 655
|
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| id | DE-604.BV006521480 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:47:38Z |
| institution | BVB |
| isbn | 030641645X |
| language | English |
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| physical | XI, 662 S. Illustrationen |
| publishDate | 1984 |
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| publisher | Plenum Press |
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| series | Mathematical concepts and methods in science and engineering |
| series2 | Mathematical concepts and methods in science and engineering |
| spelling | Golberg, Michael A. Verfasser (DE-588)1158078846 aut An introduction to probability theory with statistical applications Michael A. Golberg Probability theory New York, NY [u.a.] Plenum Press 1984 XI, 662 S. Illustrationen txt rdacontent n rdamedia nc rdacarrier Mathematical concepts and methods in science and engineering 29 Estadística matemática Mathematical statistics Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s DE-604 Mathematical concepts and methods in science and engineering 29 (DE-604)BV000001144 29 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004152393&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Golberg, Michael A. An introduction to probability theory with statistical applications Mathematical concepts and methods in science and engineering Estadística matemática Mathematical statistics Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)4151278-9 |
| title | An introduction to probability theory with statistical applications |
| title_alt | Probability theory |
| title_auth | An introduction to probability theory with statistical applications |
| title_exact_search | An introduction to probability theory with statistical applications |
| title_full | An introduction to probability theory with statistical applications Michael A. Golberg |
| title_fullStr | An introduction to probability theory with statistical applications Michael A. Golberg |
| title_full_unstemmed | An introduction to probability theory with statistical applications Michael A. Golberg |
| title_short | An introduction to probability theory with statistical applications |
| title_sort | an introduction to probability theory with statistical applications |
| topic | Estadística matemática Mathematical statistics Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Estadística matemática Mathematical statistics Probabilities Wahrscheinlichkeitsrechnung Einführung |
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