Contents
Preface vii
1 Introduction 1
1.1 The Beginnings of Probability 1
1.2 Examples of Applications 2
1.3 Relative Frequencies 4
2 Elementary Probabilities 7
2.1 Basic Concepts 7
2.2 Probability 10
2.3 Combinatorial Analysis 13
2.4 Conditional Probability 18
2.5 Independence 21
2.6 Exercises, Problems, and Multiple Choice Questions 26
3 Random Variables 55
3.1 Introduction 55
3.2 The Distribution Function 57
3.3 The Probability Mass and Density Functions 64
3.4 Important Discrete Random Variables 70
3.5 Important Continuous Random Variables 82
3.6 Transformations 92
3.7 Mathematical Expectation and Variance 95
3.8 Transforms 103
3.9 Reliability 108
3.10 Exercises, Problems, and Multiple Choice Questions Ill
4 Random Vectors 157
4.1 Introduction 157
4.2 Random Vectors of Dimension 2 158
4.3 Conditionals 166
X Contents
4.4 Random Vectors of Dimension n >2 170
4.5 Transformations of Random Vectors 172
4.6 Covariance and Correlation 176
4.7 Multinormal Distribution 179
4.8 Estimation of a Random Variable 182
4.9 Linear Combinations 185
4.10 The Laws of Large Numbers 188
4.11 The Central Limit Theorem 189
4.12 Exercises, Problems, and Multiple Choice Questions 195
5 Stochastic Processes 221
5.1 Introduction 221
5.2 Characteristics of Stochastic Processes 222
5.3 Markov Chains 225
5.4 The Poisson Process 228
5.5 The Wiener Process 232
5.6 Stationarity 235
5.7 Ergodicity 238
5.8 Exercises, Problems, and Multiple Choice Questions 240
6 Estimation and Testing 253
6.1 Point Estimation 253
6.2 Estimation by Confidence Intervals 258
6.3 (Pearson's) Chi-Square Goodness-of-Fit Test 262
6.4 Tests of Hypotheses on the Parameters 266
6.5 Exercises, Problems, and Multiple Choice Questions,
Supplementary Exercises 279
7 Simple Linear Regression 307
7.1 Introduction: The Model 307
7.2 Tests of Hypotheses 310
7.3 Confidence Intervals and Ellipses 313
7.4 The Coefficient of Determination 315
7.5 The Analysis of Residuals 315
7.6 Curvilinear Regression 318
7.7 Correlation 321
7.8 Exercises, Problems, and Multiple Choice Questions,
Supplementary Exercises 324
Appendix A: Mathematical Formulas 339
Appendix B: Quantiles of the Sampling Distributions 341
Appendix C: Classification of the Exercises 345
Appendix D: Answers to the Multiple Choice Questions 347