Introduction to Probability

Title page......Page 2
Copyright page......Page 3
Dedication......Page 4
PREFACE......Page 10
Contents......Page 6
1 SOME MOTIVATING EXAMPLES......Page 14
2.1 Some Fundamental Concepts......Page 19
2.2 Some Fundamental Results......Page 24
2.3 Random Variables......Page 32
2.4 Basic Concepts and Results in Counting......Page 36
3.1 Definition of Probability......Page 44
3.2 Some Basic Properties and Results......Page 49
3.3 Distribution of a Random Variable......Page 58
4.1 Conditional Probability and Related Results......Page 69
4.2 Independent Events and Related Results......Page 82
5.1 Expectation, Variance, and Moment-Generating Function of a Random Variable......Page 95
5.2 Some Probability Inequalities......Page 106
5.3 Median and Mode of a Random Variable......Page 109
6.1 Some Special Discrete Distributions......Page 116
6.2 Some Special Continuous Distributions......Page 134
7.1 Joint d.f. and Joint p.d.f. of Two Random Variables......Page 153
7.2 Marginal and Conditional p.d.f.’s, Conditional Expectation, and Variance......Page 165
8.1 The Joint m.g.f. of Two Random Variables......Page 180
8.2 Covariance and Correlation Coefficient of Two Random Variables......Page 185
8.3 Proof of Theorem 1, Some Further Results......Page 193
9 SOME GENERALIZATIONS TO k RANDOM VARIABLES, AND THREE MULTIVARIATE DISTRIBUTIONS......Page 198
9.1 Joint Distribution of k Random Variables and Related Quantities......Page 199
9.2 Multinomial Distribution......Page 202
9.3 Bivariate Normal Distribution......Page 210
9.4 Multivariate Normal Distribution......Page 219
10.1 Independence of Random Variables and Criteria of Independence......Page 220
10.2 The Reproductive Property of Certain Distributions......Page 233
10.3 Distribution of the Sample Variance under Normality......Page 242
11.1 Transforming a Single Random Variable......Page 245
11.2 Transforming Two or More Random Variables......Page 253
11.3 Linear Transformations......Page 268
11.4 The Probability Integral Transform......Page 278
11.5 Order Statistics......Page 280
12 TWO MODES OF CONVERGENCE, THE WEAK LAW OF LARGE NUMBERS, THE CENTRAL LIMIT THEOREM, AND FURTHER RESULTS......Page 291
12.1 Convergence in Distribution and in Probability......Page 292
12.2 The Weak Law of Large Numbers and the Central Limit Theorem......Page 298
12.3 Further Limit Theorems......Page 316
13 AN OVERVIEW OF STATISTICAL INFERENCE......Page 322
13.1 The Basics of Point Estimation......Page 323
13.2 The Basics of Interval Estimation......Page 326
13.3 The Basics of Testing Hypotheses......Page 327
13.4 The Basics of Regression Analysis......Page 331
13.5 The Basics of Analysis of Variance......Page 332
13.6 The Basics of Nonparametric Inference......Page 334
Table 1 Cumulative Binomial Distribution......Page 337
Table 2 Cumulative Poisson Distribution......Page 345
Table 3 Normal Distribution......Page 347
Table 4 Critical Values for Chi-Square Distribution......Page 350
Table 5 Table of Selected Discrete and Continuous Distributions and Some of Their Characteristics......Page 352
Table 6 Handy Reference to Some Formulas Used in the Text......Page 355
SOME NOTATIONS AND ABBREVIATIONS......Page 357
ANSWERS TO EVEN-NUMBERED EXERCISES......Page 360
INDEX......Page 394