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Summary
Summary
A thorough introduction to the fundamentals of probability theory
This book offers a detailed explanation of the basic models and mathematical principles used in applying probability theory to practical problems. It gives the reader a solid foundation for formulating and solving many kinds of probability problems for deriving additional results that may be needed in order to address more challenging questions, as well as for proceeding with the study of a wide variety of more advanced topics.
Great care is devoted to a clear and detailed development of the 'conceptual model' which serves as the bridge between any real-world situation and its analysis by means of the mathematics of probability. Throughout the book, this conceptual model is not lost sight of. Random variables in one and several dimensions are treated in detail, including singular random variables, transformations, characteristic functions, and sequences. Also included are special topics not covered in many probability texts, such as fuzziness, entropy, spherically symmetric random variables, and copulas.
Some special features of the book are:
a unique step-by-step presentation organized into 86 topical Sections, which are grouped into six Parts over 200 diagrams augment and illustrate the text, which help speed the reader's comprehension of the material short answer review questions following each Section, with an answer table provided, strengthen the reader's detailed grasp of the material contained in the Section problems associated with each Section provide practice in applying the principles discussed, and in some cases extend the scope of that material an online separate solutions manual is available for course tutors.The various features of this textbook make it possible for engineering students to become well versed in the 'machinery' of probability theory. They also make the book a useful resource for self-study by practicing engineers and researchers who need a more thorough grasp of particular topics.
Author Notes
Professor Harry Schwarzlander, Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, New York, USA
Harry Schwarzlander is Associate Professor Emeritus at Syracuse University and has been with the university since 1964 where he has developed and taught 25 courses to electrical engineering graduate and undergraduate students. He was an Instructor in the Department of Electrical Engineering at Purdue University from 1960 to 1964, and before that, an Engineer and Project Engineer for General Electronic Laboratories, Inc., Cambridge, Massachusetts.
Professor Schwarzlander is a Registered Professional Engineer in New York and a Life Member of IEEE, taking posts as Secretary and Chairman between 1967 and 1969. In 2004 he was awarded Doctor Honoris Causa 'in recognition of outstanding accomplishments, exemplary educational leadership and distinguished service to mankind' by The International Institute for Advanced Studies in Systems Research and Cybernetics. He holds one patent for the RMS-Measuring Voltmeter, 1959.
Currently Executive Director of The New Environment, Inc. and Editor of New Environment Bulletin (the monthly newsletter of the New Environment Association), Professor Schwarzlander has contributed to over 65 publications and presentations. He researches into a range of different areas, including interference testing of electronic equipment and information storage and retrieval.
Table of Contents
| Preface | p. xi |
| Introduction | p. xiii |
| Part I The Basic Model | |
| Part I Introduction | p. 2 |
| Section 1 Dealing with 'Real-World' Problems | p. 3 |
| Section 2 The Probabilistic Experiment | p. 6 |
| Section 3 Outcome | p. 11 |
| Section 4 Events | p. 14 |
| Section 5 The Connection to the Mathematical World | p. 17 |
| Section 6 Elements and Sets | p. 20 |
| Section 7 Classes of Sets | p. 23 |
| Section 8 Elementary Set Operations | p. 26 |
| Section 9 Additional Set Operations | p. 30 |
| Section 10 Functions | p. 33 |
| Section 11 The Size of a Set | p. 36 |
| Section 12 Multiple and Infinite Set Operations | p. 40 |
| Section 13 More About Additive Classes | p. 44 |
| Section 14 Additive Set Functions | p. 49 |
| Section 15 More about Probabilistic Experiments | p. 53 |
| Section 16 The Probability Function | p. 58 |
| Section 17 Probability Space | p. 62 |
| Section 18 Simple Probability Arithmetic | p. 65 |
| Part I Summary | p. 71 |
| Part II The Approach to Elementary Probability Problems | |
| Part II Introduction | p. 74 |
| Section 19 About Probability Problems | p. 75 |
| Section 20 Equally Likely Possible Outcomes | p. 81 |
| Section 21 Conditional Probability | p. 86 |
| Section 22 Conditional Probability Distributions | p. 91 |
| Section 23 Independent Events | p. 99 |
| Section 24 Classes of Independent Events | p. 104 |
| Section 25 Possible Outcomes Represented as Ordered k-Tuples | p. 109 |
| Section 26 Product Experiments and Product Spaces | p. 114 |
| Section 27 Product Probability Spaces | p. 120 |
| Section 28 Dependence Between the Components in an Ordered k-Tuple | p. 125 |
| Section 29 Multiple Observations Without Regard to Order | p. 128 |
| Section 30 Unordered Sampling with Replacement | p. 132 |
| Section 31 More Complicated Discrete Probability Problems | p. 135 |
| Section 32 Uncertainty and Randomness | p. 140 |
| Section 33 Fuzziness | p. 146 |
| Part II Summary | p. 152 |
| Part III Introduction to Random Variables | |
| Part III Introduction | p. 154 |
| Section 34 Numerical-Valued Outcomes | p. 155 |
| Section 35 The Binomial Distribution | p. 161 |
| Section 36 The Real Numbers | p. 165 |
| Section 37 General Definition of a Random Variable | p. 169 |
| Section 38 The Cumulative Distribution Function | p. 173 |
| Section 39 The Probability Density Function | p. 180 |
| Section 40 The Gaussian Distribution | p. 186 |
| Section 41 Two Discrete Random Variables | p. 191 |
| Section 42 Two Arbitrary Random Variables | p. 197 |
| Section 43 Two-Dimensional Distribution Functions | p. 202 |
| Section 44 Two-Dimensional Density Functions | p. 208 |
| Section 45 Two Statistically Independent Random Variables | p. 216 |
| Section 46 Two Statistically Independent Random Variables-Absolutely Continuous Case | p. 221 |
| Part III Summary | p. 226 |
| Part IV Transformations and Multiple Random Variables | |
| Part IV Introduction | p. 228 |
| Section 47 Transformation of a Random Variable | p. 229 |
| a Transformation of a discrete random variable | p. 229 |
| b Transformation of an arbitrary random variable | p. 231 |
| c Transformation of an absolutely continuous random variable | p. 235 |
| Section 48 Transformation of a Two-Dimensional Random Variable | p. 238 |
| Section 49 The Sum of Two Discrete Random Variables | p. 243 |
| Section 50 The Sum of Two Arbitrary Random Variables | p. 247 |
| Section 51 n-Dimensional Random Variables | p. 253 |
| Section 52 Absolutely Continuous n-Dimensional R.V.'s | p. 259 |
| Section 53 Coordinate Transformations | p. 263 |
| Section 54 Rotations and the Bivariate Gaussian Distribution | p. 268 |
| Section 55 Several Statistically Independent Random Variables | p. 274 |
| Section 56 Singular Distributions in One Dimension | p. 279 |
| Section 57 Conditional Induced Distribution, Given an Event | p. 284 |
| Section 58 Resolving a Distribution into Components of Pure Type | p. 290 |
| Section 59 Conditional Distribution Given the Value of a Random Variable | p. 293 |
| Section 60 Random Occurrences in Time | p. 298 |
| Part IV Summary | p. 304 |
| Part V Parameters for Describing Random Variables and Induced Distributions | |
| Part V Introduction | p. 306 |
| Section 61 Some Properties of a Random Variable | p. 307 |
| Section 62 Higher Moments | p. 314 |
| Section 63 Expectation of a Function of a Random Variable | p. 320 |
| a Scale change and shift of origin | p. 320 |
| b General formulation | p. 320 |
| c Sum of random variables | p. 322 |
| d Powers of a random variable | p. 323 |
| e Product of random variables | p. 325 |
| Section 64 The Variance of a Function of a Random Variable | p. 328 |
| Section 65 Bounds on the Induced Distribution | p. 332 |
| Section 66 Test Sampling | p. 336 |
| a A Simple random sample | p. 336 |
| b Unbiased estimators | p. 338 |
| c Variance of the sample average | p. 339 |
| d Estimating the population variance | p. 341 |
| e Sampling with replacement | p. 342 |
| Section 67 Conditional Expectation with Respect to an Event | p. 345 |
| Section 68 Covariance and Correlation Coefficient | p. 350 |
| Section 69 The Correlation Coefficient as Parameter in a Joint Distribution | p. 356 |
| Section 70 More General Kinds of Dependence Between Random Variables | p. 362 |
| Section 71 The Covariance Matrix | p. 367 |
| Section 72 Random Variables as the Elements of a Vector Space | p. 374 |
| Section 73 Estimation | p. 379 |
| a The concept of estimating a random variable | p. 379 |
| b Optimum constant estimates | p. 379 |
| c Mean-square estimation using random variables | p. 381 |
| d Linear mean-square estimation | p. 382 |
| Section 74 The Stieltjes Integral | p. 386 |
| Part V Summary | p. 393 |
| Part VI Further Topics in Random Variables | |
| Part VI Introduction | p. 396 |
| Section 75 Complex Random Variables | p. 397 |
| Section 76 The Characteristic Function | p. 402 |
| Section 77 Characteristic Function of a Transformed Random Variable | p. 408 |
| Section 78 Characteristic Function of a Multidimensional Random Variable | p. 412 |
| Section 79 The Generating Function | p. 417 |
| Section 80 Several Jointly Gaussian Random Variables | p. 422 |
| Section 81 Spherically Symmetric Vector Random Variables | p. 428 |
| Section 82 Entropy Associated with Random Variables | p. 435 |
| a Discrete random variables | p. 435 |
| b Absolutely continuous random variables | p. 438 |
| Section 83 Copulas | p. 443 |
| Section 84 Sequences of Random Variables | p. 454 |
| a Preliminaries | p. 454 |
| b Simple gambling schemes | p. 455 |
| c Operations on sequences | p. 458 |
| Section 85 Convergent Sequences and Laws of Large Numbers | p. 461 |
| a Convergence of sequences | p. 461 |
| b Laws of large numbers | p. 464 |
| c Connection with statistical regularity | p. 468 |
| Section 86 Convergence of Probability Distributions and the Central Limit Theorem | p. 470 |
| Part VI Summary | p. 477 |
| Appendices | p. 479 |
| Answers to Queries | p. 479 |
| Table of the Gaussian Integral | p. 482 |
| Part I Problems | p. 483 |
| Part II Problems | p. 500 |
| Part III Problems | p. 521 |
| Part IV Problems | p. 537 |
| Part V Problems | p. 556 |
| Part VI Problems | p. 574 |
| Notation and Abbreviations | p. 587 |
| References | p. 595 |
| Subject Index | p. 597 |
