Preface for the First Edition xi
Preface for the Second Edition xv
1. Sample Spaces and Probability 1
1.1. Discrete Sample Spaces 1
1.2. Events; Axioms of Probability 7
Axioms of Probability 8
1.3. Probability Theorems 10
1.4. Conditional Probability and Independence 14
Independence 23
1.5. Some Examples 28
1.6. Reliability of Systems 34
Series Systems 34
Parallel Systems 35
1.7. Counting Techniques 39
Chapter Review 54
Problems for Review 56
Supplementary Exercises for Chapter 1 56
2. Discrete Random Variables and Probability Distributions 61
2.1. Random Variables 61
2.2. Distribution Functions 68
2.3. Expected Values of Discrete Random Variables 72
Expected Value of a Discrete Random Variable 72
Variance of a Random Variable 75
Tchebycheff's Inequality 78
2.4. Binomial Distribution 81
2.5. A Recursion 82
The Mean and Variance of the Binomial 84
2.6. Some Statistical Considerations 88
2.7. Hypothesis Testing: Binomial Random Variables 92
2.8. Distribution of A Sample Proportion 98
2.9. Geometric and Negative Binomial Distributions 102
A Recursion 108
2.10. The Hypergeometric Random Variable: Acceptance Sampling 111
Acceptance Sampling 111
The Hypergeometric Random Variable 114
Some Specific Hypergeometric Distributions 116
2.11. Acceptance Sampling (Continued) 119
Producer's and Consumer's Risks 121
Average Outgoing Quality 122
Double Sampling 124
2.12. The Hypergeometric Random Variable: Further Examples 128
2.13. The Poisson Random Variable 130
Mean and Variance of the Poisson 131
Some Comparisons 132
2.14. The Poisson Process 134
Chapter Review 139
Problems for Review 141
Supplementary Exercises for Chapter 2 142
3. Continuous Random Variables and Probability Distributions 146
3.1. Introduction 146
Mean and Variance 150
A Word on Words 153
3.2. Uniform Distribution 157
3.3. Exponential Distribution 159
Mean and Variance 160
Distribution Function 161
3.4. Reliability 162
Hazard Rate 163
3.5. Normal Distribution 166
3.6. Normal Approximation to the Binomial Distribution 175
3.7. Gamma and Chi-Squared Distributions 178
3.8. Weibull Distribution 184
Chapter Review 186
Problems For Review 189
Supplementary Exercises for Chapter 3 189
4. Functions of Random Variables; Generating Functions; Statistical Applications 194
4.1. Introduction 194
4.2. Some Examples of Functions of Random Variables 195
4.3. Probability Distributions of Functions of Random Variables 196
Expectation of a Function of X 199
4.4. Sums of Random Variables I 203
4.5. Generating Functions 207
4.6. Some Properties of Generating Functions 211
4.7. Probability Generating Functions for Some Specific Probability Distributions 213
Binomial Distribution 213
Poisson's Trials 214
Geometric Distribution 215
Collecting Premiums in Cereal Boxes 216
4.8. Moment Generating Functions 218
4.9. Properties of Moment Generating Functions 223
4.10. Sums of Random Variables-II 224
4.11. The Central Limit Theorem 229
4.12. Weak Law of Large Numbers 233
4.13. Sampling Distribution of the Sample Variance 234
4.14. Hypothesis Tests and Confidence Intervals for a Single Mean 240
Confidence Intervals, 𝜎 Known 241
Student's t Distribution 242
p Values 243
4.15. Hypothesis Tests on Two Samples 248
Tests on Two Means 248
Tests on Two Variances 251
4.16. Least Squares Linear Regression 258
4.17. Quality Control Chart for X 266
Chapter Review 271
Problems for Review 275
Supplementary Exercises for Chapter 4 275
5. Bivariate Probability Distributions 283
5.1. Introduction 283
5.2. Joint and Marginal Distributions 283
5.3. Conditional Distributions and Densities 293
5.4. Expected Values and the Correlation Coefficient 298
5.5. Conditional Expectations 303
5.6. Bivariate Normal Densities 308
Contour Plots 310
5.7. Functions of Random Variables 312
Chapter Review 316
Problems for Review 317
Supplementary Exercises for Chapter 5 317
6. Recursions and Markov Chains 322
6.1. Introduction 322
6.2. Some Recursions and their Solutions 322
Solution of the Recursion (6.3) 326
Mean and Variance 329
6.3. Random Walk and Ruin 334
Expected Duration of the Game 337
6.4. Waiting Times for Patterns in Bernoulli Trials 339
Generating Functions 341
Average Waiting Times 342
Means and Variances by Generating Functions 343
6.5. Markov Chains 344
Chapter Review 354
Problems for Review 355
Supplementary Exercises for Chapter 6 355
7. Some Challenging Problems 357
7.1. My Socks and 𝜋 357
7.2. Expected Value 359
7.3. Variance 361
7.4. Other Socks Problems 362
7.5. Coupon Collection and Related Problems 362
Three Prizes 363
Permutations 363
An Alternative Approach 363
Altering the Probabilities 364
A General Result 364
Expectations and Variances 366
Geometric Distribution 366
Variances 367
Waiting for Each of the Integers 367
Conditional Expectations 368
Other Expected Values 369
Waiting for All the Sums on Two Dice 370
7.6. Conclusion 372
7.7. Jackknifed Regression and the Bootstrap 372
Jackknifed Regression 372
7.8. Cook's Distance 374
7.9. The Bootstrap 375
7.10. On Waldegrave's Problem 378
Three Players 378
7.11. Probabilities of Winning 378
7.12. More than Three Players 379
r + 1 Players 381
Probabilities of Each Player 382
Expected Length of the Series 383
Fibonacci Series 383
7.13. Conclusion 384
7.14. On Huygen's First Problem 384
7.15. Changing the Sums for the Players 384
Decimal Equivalents 386
Another Order 387
Bernoulli's Sequence 387
Bibliography 388
Appendix A. Use of Mathematica in Probability and Statistics 390
Appendix B. Answers for Odd-Numbered Exercises 429
Appendix C. Standard Normal Distribution 453
Index 461
