Probability Concepts and Theory for Engineers Harry Schwarzlander, Formerly Department of Electrical and Computer Engineering, Syracuse University, Syracuse, NY, USA 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…mehr
Probability Concepts and Theory for Engineers Harry Schwarzlander, Formerly Department of Electrical and Computer Engineering, Syracuse University, Syracuse, NY, USA 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. Engineering students using this text will achieve a solid understanding and confidence in applying probability theory. It is also a useful resource for self-study, and for practicing engineers and researchers who need a more thorough grasp of particular topics.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
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.
Inhaltsangabe
Preface xi Introduction xiii Part I The Basic Model Part I Introduction 2 Section 1 Dealing with 'Real-World' Problems 3 Section 2 The Probabilistic Experiment 6 Section 3 Outcome 11 Section 4 Events 14 Section 5 The Connection to the Mathematical World 17 Section 6 Elements and Sets 20 Section 7 Classes of Sets 23 Section 8 Elementary Set Operations 26 Section 9 Additional Set Operations 30 Section 10 Functions 33 Section 11 The Size of a Set 36 Section 12 Multiple and Infinite Set Operations 40 Section 13 More About Additive Classes 44 Section 14 Additive Set Functions 49 Section 15 More about Probabilistic Experiments 53 Section 16 The Probability Function 58 Section 17 Probability Space 62 Section 18 Simple Probability Arithmetic 65 Part I Summary 71 Part II The Approach to Elementary Probability Problems Part II Introduction 74 Section 19 About Probability Problems 75 Section 20 Equally Likely Possible Outcomes 81 Section 21 Conditional Probability 86 Section 22 Conditional Probability Distributions 91 Section 23 Independent Events 99 Section 24 Classes of Independent Events 104 Section 25 Possible Outcomes Represented as Ordered k-Tuples 109 Section 26 Product Experiments and Product Spaces 114 Section 27 Product Probability Spaces 120 Section 28 Dependence Between the Components in an Ordered k-Tuple 125 Section 29 Multiple Observations Without Regard to Order 128 Section 30 Unordered Sampling with Replacement 132 Section 31 More Complicated Discrete Probability Problems 135 Section 32 Uncertainty and Randomness 140 Section 33 Fuzziness 146 Part II Summary 152 Part III Introduction to Random Variables Part III Introduction 154 Section 34 Numerical-Valued Outcomes 155 Section 35 The Binomial Distribution 161 Section 36 The Real Numbers 165 Section 37 General Definition of a Random Variable 169 Section 38 The Cumulative Distribution Function 173 Section 39 The Probability Density Function 180 Section 40 The Gaussian Distribution 186 Section 41 Two Discrete Random Variables 191 Section 42 Two Arbitrary Random Variables 197 Section 43 Two-Dimensional Distribution Functions 202 Section 44 Two-Dimensional Density Functions 208 Section 45 Two Statistically Independent Random Variables 216 Section 46 Two Statistically Independent Random Variables-Absolutely Continuous Case 221 Part III Summary 226 Part IV Transformations and Multiple Random Variables Part IV Introduction 228 Section 47 Transformation of a Random Variable 229 a) Transformation of a discrete random variable 229 b) Transformation of an arbitrary random variable 231 c) Transformation of an absolutely continuous random variable 235 Section 48 Transformation of a Two-Dimensional Random Variable 238 Section 49 The Sum of Two Discrete Random Variables 243 Section 50 The Sum of Two Arbitrary Random Variables 247 Section 51 n-Dimensional Random Variables 253 Section 52 Absolutely Continuous n-Dimensional R.V.'s 259 Section 53 Coordinate Transformations 263 Section 54 Rotations and the Bivariate Gaussian Distribution 268 Section 55 Several Statistically Independent Random Variables 274 Section 56 Singular Distributions in One Dimension 279 Section 57 Conditional Induced Distribution, Given an Event 284 Section 58 Resolving a Distribution into Components of Pure Type 290 Section 59 Conditional Distribution Given the Value of a Random Variable 293 Section 60 Random Occurrences in Time 298 Part IV Summary 304 Part V Parameters for Describing Random Variables and Induced Distributions Part V Introduction 306 Section 61 Some Properties of a Random Variable 307 Section 62 Higher Moments 314 Section 63 Expectation of a Function of a Random Variable 320 a) Scale change and shift of origin 320 b) General formulation 320 c) Sum of random variables 322 d) Powers of a random variable 323 e) Product of random variables 325 Section 64 The Variance of a Function of a Random Variable 328 Section 65 Bounds on the Induced Distribution 332 Section 66 Test Sampling 336 a) A Simple random sample 336 b) Unbiased estimators 338 c) Variance of the sample average 339 d) Estimating the population variance 341 e) Sampling with replacement 342 Section 67 Conditional Expectation with Respect to an Event 345 Section 68 Covariance and Correlation Coefficient 350 Section 69 The Correlation Coefficient as Parameter in a Joint Distribution 356 Section 70 More General Kinds of Dependence Between Random Variables 362 Section 71 The Covariance Matrix 367 Section 72 Random Variables as the Elements of a Vector Space 374 Section 73 Estimation 379 a) The concept of estimating a random variable 379 b) Optimum constant estimates 379 c) Mean-square estimation using random variables 381 d) Linear mean-square estimation 382 Section 74 The Stieltjes Integral 386 Part V Summary 393 Part VI Further Topics in Random Variables Part VI Introduction 396 Section 75 Complex Random Variables 397 Section 76 The Characteristic Function 402 Section 77 Characteristic Function of a Transformed Random Variable 408 Section 78 Characteristic Function of a Multidimensional Random Variable 412 Section 79 The Generating Function 417 Section 80 Several Jointly Gaussian Random Variables 422 Section 81 Spherically Symmetric Vector Random Variables 428 Section 82 Entropy Associated with Random Variables 435 a) Discrete random variables 435 b) Absolutely continuous random variables 438 Section 83 Copulas 443 Section 84 Sequences of Random Variables 454 a) Preliminaries 454 b) Simple gambling schemes 455 c) Operations on sequences 458 Section 85 Convergent Sequences and Laws of Large Numbers 461 a) Convergence of sequences 461 b) Laws of large numbers 464 c) Connection with statistical regularity 468 Section 86 Convergence of Probability Distributions and the Central Limit Theorem 470 Part VI Summary 477 Appendices 479 Answers to Queries 479 Table of the Gaussian Integral 482 Part I Problems 483 Part II Problems 500 Part III Problems 521 Part IV Problems 537 Part V Problems 556 Part VI Problems 574 Notation and Abbreviations 587 References 595 Subject Index 597
Preface xi Introduction xiii Part I The Basic Model Part I Introduction 2 Section 1 Dealing with 'Real-World' Problems 3 Section 2 The Probabilistic Experiment 6 Section 3 Outcome 11 Section 4 Events 14 Section 5 The Connection to the Mathematical World 17 Section 6 Elements and Sets 20 Section 7 Classes of Sets 23 Section 8 Elementary Set Operations 26 Section 9 Additional Set Operations 30 Section 10 Functions 33 Section 11 The Size of a Set 36 Section 12 Multiple and Infinite Set Operations 40 Section 13 More About Additive Classes 44 Section 14 Additive Set Functions 49 Section 15 More about Probabilistic Experiments 53 Section 16 The Probability Function 58 Section 17 Probability Space 62 Section 18 Simple Probability Arithmetic 65 Part I Summary 71 Part II The Approach to Elementary Probability Problems Part II Introduction 74 Section 19 About Probability Problems 75 Section 20 Equally Likely Possible Outcomes 81 Section 21 Conditional Probability 86 Section 22 Conditional Probability Distributions 91 Section 23 Independent Events 99 Section 24 Classes of Independent Events 104 Section 25 Possible Outcomes Represented as Ordered k-Tuples 109 Section 26 Product Experiments and Product Spaces 114 Section 27 Product Probability Spaces 120 Section 28 Dependence Between the Components in an Ordered k-Tuple 125 Section 29 Multiple Observations Without Regard to Order 128 Section 30 Unordered Sampling with Replacement 132 Section 31 More Complicated Discrete Probability Problems 135 Section 32 Uncertainty and Randomness 140 Section 33 Fuzziness 146 Part II Summary 152 Part III Introduction to Random Variables Part III Introduction 154 Section 34 Numerical-Valued Outcomes 155 Section 35 The Binomial Distribution 161 Section 36 The Real Numbers 165 Section 37 General Definition of a Random Variable 169 Section 38 The Cumulative Distribution Function 173 Section 39 The Probability Density Function 180 Section 40 The Gaussian Distribution 186 Section 41 Two Discrete Random Variables 191 Section 42 Two Arbitrary Random Variables 197 Section 43 Two-Dimensional Distribution Functions 202 Section 44 Two-Dimensional Density Functions 208 Section 45 Two Statistically Independent Random Variables 216 Section 46 Two Statistically Independent Random Variables-Absolutely Continuous Case 221 Part III Summary 226 Part IV Transformations and Multiple Random Variables Part IV Introduction 228 Section 47 Transformation of a Random Variable 229 a) Transformation of a discrete random variable 229 b) Transformation of an arbitrary random variable 231 c) Transformation of an absolutely continuous random variable 235 Section 48 Transformation of a Two-Dimensional Random Variable 238 Section 49 The Sum of Two Discrete Random Variables 243 Section 50 The Sum of Two Arbitrary Random Variables 247 Section 51 n-Dimensional Random Variables 253 Section 52 Absolutely Continuous n-Dimensional R.V.'s 259 Section 53 Coordinate Transformations 263 Section 54 Rotations and the Bivariate Gaussian Distribution 268 Section 55 Several Statistically Independent Random Variables 274 Section 56 Singular Distributions in One Dimension 279 Section 57 Conditional Induced Distribution, Given an Event 284 Section 58 Resolving a Distribution into Components of Pure Type 290 Section 59 Conditional Distribution Given the Value of a Random Variable 293 Section 60 Random Occurrences in Time 298 Part IV Summary 304 Part V Parameters for Describing Random Variables and Induced Distributions Part V Introduction 306 Section 61 Some Properties of a Random Variable 307 Section 62 Higher Moments 314 Section 63 Expectation of a Function of a Random Variable 320 a) Scale change and shift of origin 320 b) General formulation 320 c) Sum of random variables 322 d) Powers of a random variable 323 e) Product of random variables 325 Section 64 The Variance of a Function of a Random Variable 328 Section 65 Bounds on the Induced Distribution 332 Section 66 Test Sampling 336 a) A Simple random sample 336 b) Unbiased estimators 338 c) Variance of the sample average 339 d) Estimating the population variance 341 e) Sampling with replacement 342 Section 67 Conditional Expectation with Respect to an Event 345 Section 68 Covariance and Correlation Coefficient 350 Section 69 The Correlation Coefficient as Parameter in a Joint Distribution 356 Section 70 More General Kinds of Dependence Between Random Variables 362 Section 71 The Covariance Matrix 367 Section 72 Random Variables as the Elements of a Vector Space 374 Section 73 Estimation 379 a) The concept of estimating a random variable 379 b) Optimum constant estimates 379 c) Mean-square estimation using random variables 381 d) Linear mean-square estimation 382 Section 74 The Stieltjes Integral 386 Part V Summary 393 Part VI Further Topics in Random Variables Part VI Introduction 396 Section 75 Complex Random Variables 397 Section 76 The Characteristic Function 402 Section 77 Characteristic Function of a Transformed Random Variable 408 Section 78 Characteristic Function of a Multidimensional Random Variable 412 Section 79 The Generating Function 417 Section 80 Several Jointly Gaussian Random Variables 422 Section 81 Spherically Symmetric Vector Random Variables 428 Section 82 Entropy Associated with Random Variables 435 a) Discrete random variables 435 b) Absolutely continuous random variables 438 Section 83 Copulas 443 Section 84 Sequences of Random Variables 454 a) Preliminaries 454 b) Simple gambling schemes 455 c) Operations on sequences 458 Section 85 Convergent Sequences and Laws of Large Numbers 461 a) Convergence of sequences 461 b) Laws of large numbers 464 c) Connection with statistical regularity 468 Section 86 Convergence of Probability Distributions and the Central Limit Theorem 470 Part VI Summary 477 Appendices 479 Answers to Queries 479 Table of the Gaussian Integral 482 Part I Problems 483 Part II Problems 500 Part III Problems 521 Part IV Problems 537 Part V Problems 556 Part VI Problems 574 Notation and Abbreviations 587 References 595 Subject Index 597
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