Pfeiffer / Schum Introduction to Applied Probability

1. Auflage 2014
ISBN: 978-1-4832-7720-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 420 Seiten, Web PDF

Introduction to Applied Probability provides a basis for an intelligent application of probability ideas to a wide variety of phenomena for which it is suitable. It is intended as a tool for learning and seeks to point out and emphasize significant facts and interpretations which are frequently overlooked or confused by the beginner. The book covers more than enough material for a one semester course, enhancing the value of the book as a reference for the student. Notable features of the book are: the systematic handling of combinations of events (Section 3-5); extensive use of the mass concept as an aid to visualization; an unusually careful treatment of conditional probability, independence, and conditional independence (Section 6-4); the resulting clarification facilitates the formulation of many applied problems; the emphasis on events determined by random variables, which gives unity and clarity to many topics important for interpretation; and the utilization of the indicator function, both as a tool for dealing with events and as a notational device in the handling of random variables. Students of mathematics, engineering, biological and physical sciences will find the text highly useful.

Autoren/Hrsg.

Pfeiffer, Paul E.

Schum, David A.

Inhaltsverzeichnis

1;Front Cover;1
2;Introduction to Applied Probability;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;12
6;Acknowledgments;16
7;Part I. INTRODUCTION;18
7.1;Chapter 1. An Approach to Probability;20
7.1.1;INTRODUCTION;20
7.1.2;1-1. Classical Probability;21
7.1.3;1-2. Toward a More General Theory;23
7.2;Chapter 2. Some Elementary Strategies of Counting;27
7.2.1;Introduction;27
7.2.2;2-1. Basic Principles;28
7.2.3;2-2. Arrangements;31
7.2.4;2-3. Binomial Coefficients;35
7.2.5;2-4. Verification of the Formulas for Arrangements;37
7.2.6;2-5. A Formal Representation of the Arrangement Problem;40
7.2.7;2-6. An Occupancy Problem Equivalent to the Arrangement Problem;42
7.2.8;2-7. Some Problems Utilizing Elementary Arrangements and Occupancy Situations as Component Operations;43
7.2.9;Problems;52
8;PART II . BASIC PROBABILITY MODEL;54
8.1;Chapter 3. Sets and Events;56
8.1.1;Introduction;56
8.1.2;3-1. A Well-Defined Trial and Its Possible Outcomes;57
8.1.3;3-2. Events and the Occurrence of Events;59
8.1.4;3-3. Special Events and Compound Events;62
8.1.5;3-4. Classes of Events;65
8.1.6;3-5. Techniques for Handling Events;66
8.1.7;Problems;81
8.2;Chapter 4. A Probability System;84
8.2.1;Introduction;84
8.2.2;4-1. Requirements for a Formal Probability System;85
8.2.3;4-2. Basic Properties of a Probability System;87
8.2.4;4-3. Derived Properties of the Probability System;88
8.2.5;4-4. A Physical Analogy: Probability as Mass;91
8.2.6;4-5. Probability Mass Assignment on a Discrete Basic Space;92
8.2.7;4-6. On the Determination of Probabilities;94
8.2.8;4-7. Supplementary Examples;97
8.2.9;Problems;99
8.3;Chapter 5. Conditional Probability;101
8.3.1;Introduction;101
8.3.2;5-1. Conditioning and the Assignment of Probabilities;102
8.3.3;5-2. Some Properties of Conditional Probability;107
8.3.4;5-3. Supplementary Examples;115
8.3.5;5-4. Repeated Conditioning;119
8.3.6;5-5. Some Patterns of Inference;121
8.3.7;Problems;127
8.4;Chapter 6. Independence in Probability Theory;130
8.4.1;Introduction;130
8.4.2;6-1. The Defining Condition;130
8.4.3;6-2. Some Elementary Properties;132
8.4.4;6-3. Independent Classes of Events;133
8.4.5;6-4. Conditional Independence;140
8.4.6;6-5. Supplementary Examples;145
8.4.7;Problems;151
8.5;Chapter 7. Composite Trials and Sequences of Events;154
8.5.1;Introduction;154
8.5.2;7-1. Composite Trials;155
8.5.3;7-2. Repeated Trials;157
8.5.4;7-3. Bernoulli Trials;162
8.5.5;7-4. Sequences of Events;165
8.5.6;Problems;172
9;PART III . RANDOM VARIABLES;176
9.1;Chapter 8. Random Variables;178
9.1.1;Introduction;178
9.1.2;8-1. The Random Variable as a Function;178
9.1.3;8-2. Functions as Mappings;181
9.1.4;8-3. Events Determined by a Random Variable;183
9.1.5;8-4. The Indicator Function;185
9.1.6;8-5. Discrete Random Variables;189
9.1.7;8-6. Mappings and Inverse Images for Simple Random Variables;194
9.1.8;8-7. Mappings and Mass Transfer;195
9.1.9;8-8. Approximation by Simple Random Variables;198
9.1.10;Problems;199
9.2;Chapter 9. Distribution and Density Functions;202
9.2.1;Introduction;202
9.2.2;9-1. Some Introductory Examples;202
9.2.3;9-2. The Probability Distribution Function;206
9.2.4;9-3. Probability Mass and Density Functions;211
9.2.5;9-4. Additional Examples of Probability Mass Distributions;215
9.2.6;Problems;221
9.3;Chapter 10. Joint Probability Distributions;224
9.3.1;Introduction;224
9.3.2;10-1. Joint Mappings;225
9.3.3;10-2. Joint Distributions;229
9.3.4;10-3. Marginal Distributions;231
9.3.5;10-4. Properties of Joint Distribution Functions;235
9.3.6;10-5. Mass and Density Functions;237
9.3.7;10-6. Mixed Distributions;239
9.3.8;Problems;240
9.4;Chapter 11 Independence of Random Variables;242
9.4.1;Introduction;242
9.4.2;11-1. Definition and Examples;242
9.4.3;11-2. Independence and Probability Mass Distributions;244
9.4.4;11-3. A Simpler Condition for Independence;246
9.4.5;11-4. Independence Conditions for Distribution and density Functions;248
9.4.6;Problems;251
9.5;Chapter 12. Functions of Random Variables;254
9.5.1;Introduction;254
9.5.2;12-1. Examples and Definition;254
9.5.3;12-2. Distribution and Mapping for a Function of a Single Random Variable;255
9.5.4;12-3. Functions of Two Random Variables;261
9.5.5;12-4. Independence of Functions of Random Variables;266
9.5.6;Problems;268
10;PART IV. MATHEMATICAL EXPECTATION;272
10.1;Chapter 13. Mathematical Expectation and Mean Value;274
10.1.1;Introduction;274
10.1.2;13-1. The Concept;275
10.1.3;13-2. Fundamental Formulas;277
10.1.4;13-3. A Mechanical Interpretation;278
10.1.5;13-4. The Mean Value;279
10.1.6;13-5. Some General Properties;283
10.1.7;Problems;289
10.2;Chapter 14. Variance and Other Movements;291
10.2.1;Introduction;291
10.2.2;14-1. Definition and Interpretation of Variance;291
10.2.3;14-2. Some Properties of Variance;292
10.2.4;14-3. Variance for Some Common Distributions;293
10.2.5;14-4. Other Moments;297
10.2.6;14-5. Moment-Generating Function and Characteristic Function;300
10.2.7;14-6. Some Common Distributions;305
10.2.8;Problems;310
10.3;Chapter 15. Correlation and Covariance;313
10.3.1;Introduction;313
10.3.2;15-1. Joint Distributions for Centered and Standardized Random Variables;314
10.3.3;15-2. Characterization of the Joint Distributions;316
10.3.4;15-3. Covariance and the Correlation Coefficient;319
10.3.5;15-4. Linear Regression;321
10.3.6;15-5. Additional Interpretations of p;323
10.3.7;15-6. Linear Transformations of Uncorrelated Random Variables;325
10.3.8;Problems;325
10.4;Chapter 16. Conditional Expectation;328
10.4.1;Introduction;328
10.4.2;16-1. Averaging Over a Conditioning Event;328
10.4.3;16-2. A Conditioning Event Determined by a Second Random Variable;331
10.4.4;16-3. Averaging Over a Partition of an Event;331
10.4.5;16-4. Conditioning by a Discrete Random Variable;332
10.4.6;16-5. Conditioning by a Continuous Random Variable;334
10.4.7;16-6. Some Properties of Conditional Expectation;337
10.4.8;16-7. Regression Theory;339
10.4.9;16-8. Estimating a Probability;341
10.4.10;Problems;343
11;PART V. SEQUENCES OF RANDOM VARIABLES;346
11.1;Chapter 17. Sequences of Random Variables;348
11.1.1;Introduction;348
11.1.2;17-1. Composite Trials;349
11.1.3;17-2. The Multinomial Distribution;350
11.1.4;17-3. The Law of Large Numbers;351
11.1.5;17-4. The Strong Law of Large Numbers;354
11.1.6;17-5. The Central Limit Theorem;357
11.1.7;17-6. Applications to Statistics;359
11.1.8;Problems;364
11.2;Chapter 18. Constant Markov Chains;366
11.2.1;Introduction;366
11.2.2;18-1. Definitions and an Introductory Example;366
11.2.3;18-2. Some Examples of Markov Chains;371
11.2.4;18-3. Transition Diagrams and Accessibility of States;376
11.2.5;18-4. Recurrence and Periodicity;383
11.2.6;18-5. Some Results for Irreducible Chains;386
11.2.7;Problems;387
11.3;Appendix A. Numerical Tables;390
11.3.1;A-l. Factorials and Their Logarithms;390
11.3.2;A-2. The Exponential Function;390
11.3.3;A-3. Binomial Coefficients;390
11.3.4;A-4. The Summed Binomial Distribution;390
11.3.5;A-5. Standardized Normal Distribution Function;390
11.4;Appendix B. Some Mathematical Aids;398
11.4.1;B-l. Binary Representation of Numbers;398
11.4.2;B-2. Geometric Series;399
11.4.3;B-3. Extended Binomial Coefficient;399
11.4.4;B-4. Gamma Function;400
11.4.5;B-5. Beta Function;400
11.4.6;B-6. Matrices;401
11.5;Selected References;404
11.6;Selected Answers and Hints;406
11.7;Index of Symbols and Abbreviations;412
11.8;Index;414