Probability: An Introduction with Statistical Applications, 2nd Edition

**Praise for the __First Edition__****"This is a well-written and impressively presented introduction to probability and statistics. The text throughout is highly readable, and the author makes liberal use of graphs and diagrams to clarify the theory." - __The Statistician__**Thoroughly updated, __Probability: An Introduction with Statistical Applications, Second Edition__ features a comprehensive exploration of statistical data analysis as an application of probability. The new edition provides an introduction to statistics with accessible coverage of reliability, acceptance sampling, confidence intervals, hypothesis testing, and simple linear regression. Encouraging readers to develop a deeper intuitive understanding of probability, the author presents illustrative geometrical presentations and arguments without the need for rigorous mathematical proofs. The __Second Edition__ features interesting and practical examples from a variety of engineering and scientific fields, as well as: * Over 880 problems at varying degrees of difficulty allowing readers to take on more challenging problems as their skill levels increase * Chapter-by-chapter projects that aid in the visualization of probability distributions * New coverage of statistical quality control and quality production * An appendix dedicated to the use of Mathematica® and a companion website containing the referenced data sets Featuring a practical and real-world approach, this textbook is ideal for a first course in probability for students majoring in statistics, engineering, business, psychology, operations research, and mathematics. __Probability: An Introduction with Statistical Applications, Second Edition__ is also an excellent reference for researchers and professionals in any discipline who need to make decisions based on data as well as readers interested in learning how to accomplish effective decision making from data. Cover 1 Title Page 5 Copyright 6 Dedication 7 Contents 9 Preface for the First Edition 13 Preface for the Second Edition 17 Chapter 1 Sample Spaces and Probability 19 1.1. Discrete Sample Spaces 19 1.2. Events; Axioms of Probability 25 Axioms of Probability 26 1.3. Probability Theorems 28 1.4. Conditional Probability and Independence 32 Independence 41 1.5. Some Examples 46 1.6. Reliability of Systems 52 Series Systems 52 Parallel Systems 53 1.7. Counting Techniques 57 Chapter Review 72 Problems for Review 74 Supplementary Exercises for Chapter 1 74 Chapter 2 Discrete Random Variables and Probability Distributions 79 2.1. Random Variables 79 2.2. Distribution Functions 86 2.3. Expected Values of Discrete Random Variables 90 Expected Value of a Discrete Random Variable 90 Variance of a Random Variable 93 Tchebycheff's Inequality 96 2.4. Binomial Distribution 99 2.5. A Recursion 100 The Mean and Variance of the Binomial 102 2.6. Some Statistical Considerations 106 2.7. Hypothesis Testing: Binomial Random Variables 110 2.8. Distribution of A Sample Proportion 116 2.9. Geometric and Negative Binomial Distributions 120 A Recursion 126 2.10. The Hypergeometric Random Variable: Acceptance Sampling 129 Acceptance Sampling 129 The Hypergeometric Random Variable 132 Some Specific Hypergeometric Distributions 134 2.11. Acceptance Sampling (Continued) 137 Producer's and Consumer's Risks 139 Average Outgoing Quality 140 Double Sampling 142 2.12. The Hypergeometric Random Variable: Further Examples 146 2.13. The Poisson Random Variable 148 Mean and Variance of the Poisson 149 Some Comparisons 150 2.14. The Poisson Process 152 Chapter Review 157 Problems for Review 159 Supplementary Exercises for Chapter 2 160 Chapter 3 Continuous Random Variables and Probability Distributions 164 3.1. Introduction 164 Mean and Variance 168 A Word on Words 171 3.2. Uniform Distribution 175 3.3. Exponential Distribution 177 Mean and Variance 178 Distribution Function 179 3.4. Reliability 180 Hazard Rate 181 3.5. Normal Distribution 184 3.6. Normal Approximation to the Binomial Distribution 193 3.7. Gamma and Chi-Squared Distributions 196 3.8. Weibull Distribution 202 Chapter Review 204 Problems For Review 207 Supplementary Exercises for Chapter 3 207 Chapter 4 Functions of Random Variables; Generating Functions; Statistical Applications 212 4.1. Introduction 212 4.2. Some Examples of Functions of Random Variables 213 4.3. Probability Distributions of Functions of Random Variables 214 Expectation of a Function of X 217 4.4. Sums of Random Variables I 221 4.5. Generating Functions 225 4.6. Some Properties of Generating Functions 229 4.7. Probability Generating Functions for Some Specific Probability Distributions 231 Binomial Distribution 231 Poisson's Trials 232 Geometric Distribution 233 Collecting Premiums in Cereal Boxes 234 4.8. Moment Generating Functions 236 4.9. Properties of Moment Generating Functions 241 4.10. Sums of Random Variables-II 242 4.11. The Central Limit Theorem 247 4.12. Weak Law of Large Numbers 251 4.13. Sampling Distribution of the Sample Variance 252 4.14. Hypothesis Tests and Confidence Intervals for a Single Mean 258 Confidence Intervals, \sigma Known 259 Student's t Distribution 260 p Values 261 4.15. Hypothesis Tests on Two Samples 266 Tests on Two Means 266 Tests on Two Variances 269 4.16. Least Squares Linear Regression 276 4.17. Quality Control Chart for X 284 Chapter Review 289 Problems for Review 293 Supplementary Exercises for Chapter 4 293 Chapter 5 Bivariate Probability Distributions 301 5.1. Introduction 301 5.2. Joint and Marginal Distributions 301 5.3. Conditional Distributions and Densities 311 5.4. Expected Values and the Correlation Coefficient 316 5.5. Conditional Expectations 321 5.6. Bivariate Normal Densities 326 Contour Plots 328 5.7. Functions of Random Variables 330 Chapter Review 334 Problems for Review 335 Supplementary Exercises for Chapter 5 335 Chapter 6 Recursions and Markov Chains 340 6.1. Introduction 340 6.2. Some Recursions and their Solutions 340 Solution of the Recursion (6.3) 344 Mean and Variance 347 6.3. Random Walk and Ruin 352 Expected Duration of the Game 355 6.4. Waiting Times for Patterns in Bernoulli Trials 357 Generating Functions 359 Average Waiting Times 360 Means and Variances by Generating Functions 361 6.5. Markov Chains 362 Chapter Review 372 Problems for Review 373 Supplementary Exercises for Chapter 6 373 Chapter 7 Some Challenging Problems 375 7.1. My Socks and \pi 375 7.2. Expected Value 377 7.3. Variance 379 7.4. Other ``Socks'' Problems 380 7.5. Coupon Collection and Related Problems 380 Three Prizes 381 Permutations 381 An Alternative Approach 381 Altering the Probabilities 382 A General Result 382 Expectations and Variances 384 Geometric Distribution 384 Variances 385 Waiting for Each of the Integers 385 Conditional Expectations 386 Other Expected Values 387 Waiting for All the Sums on Two Dice 388 7.6. Conclusion 390 7.7. Jackknifed Regression and the Bootstrap 390 Jackknifed Regression 390 7.8. Cook's Distance 392 7.9. The Bootstrap 393 7.10. On Waldegrave's Problem 396 Three Players 396 7.11. Probabilities of Winning 396 7.12. More than Three Players 397 r + 1 Players 399 Probabilities of Each Player 400 Expected Length of the Series 401 Fibonacci Series 401 7.13. Conclusion 402 7.14. On Huygen's First Problem 402 7.15. Changing the Sums for the Players 402 Decimal Equivalents 404 Another order 405 Bernoulli's Sequence 405 Bibliography 406 Appendix A Use of Mathematica in Probability and Statistics 418 Appendix B Answers for Odd-Numbered Exercises 447 Appendix C Standard Normal Distribution 471 Index 479 End User License Agreement 483 Praise for the First Edition "This is a well-written and impressively presented introduction to probability and statistics. The text throughout is highly readable, and the author makes liberal use of graphs and diagrams to clarify the theory." - The Statistician Thoroughly updated, Probability: An Introduction with Statistical Applications, Second Edition features a comprehensive exploration of statistical data analysis as an application of probability. The new edition provides an introduction to statistics with accessible coverage of reliability, acceptance sampling, confidence intervals, hypothesis testing, and simple linear regression. Encouraging readers to develop a deeper intuitive understanding of probability, the author presents illustrative geometrical presentations and arguments without the need for rigorous mathematical proofs. The Second Edition features interesting and practical examples from a variety of engineering and scientific fields, as well as: Over 880 problems at varying degrees of difficulty allowing readers to take on more challenging problems as their skill levels increase Chapter-by-chapter projects that aid in the visualization of probability distributions New coverage of statistical quality control and quality production An appendix dedicated to the use of Mathematica ® and a companion website containing the referenced data sets Featuring a practical and real-world approach, this textbook is ideal for a first course in probability for students majoring in statistics, engineering, business, psychology, operations research, and mathematics. Probability: An Introduction with Statistical Applications, Second Edition is also an excellent reference for researchers and professionals in any discipline who need to make decisions based on data as well as readers interested in learning how to accomplish effective decision making from data.