Table of Contents
Preface to the Second Edition v
Preface to the First Edition vii
1 A First Glimpse of Probability 1
1.1 What Is Probability? 1
1.2 Experiments: Deterministic or Random 3
1.3 The Role of Probability in Statistical Inference 3
1.4 Interpreting Probability 4
Problems 6
2 Basic Concepts of Probability 9
2.1 Sample Space 9
2.2 Events and Their Probabilities 10
2.3 Combining Events 12
2.4 Probabilities Associated with Combined Events 14
2.5 Finding Probabilities 20
Problems 23
3 Counting Procedures and their Applications in Computing Probabilities 31
3.1 The Need for Counting Techniques: The Uniform Model 31
3.2 Counting Procedures Involving Order Restrictions 33
3.3 Counting Procedures Not Involving Order Restrictions 36
3.4 Applications of Counting Procedures 37
3.5 Occupancy Problems: Role of Distinguishability 43
3.6 Random Sampling 46
Problems 51
4 Conditional Probability 59
4.1 Reduction of the Sample Space 59
4.2 Multiplication Rule and Assigning Probabilities 65
4.3 Stagewise Experiments 67
4.4 Posterior Probabilities: Bayes' Rule 70
Problems 74
5 Independence 81
5.1 Independence of Two Events 81
5.2 Independence for More than Two Events 85
5.3 Probabilities Associated with Mutually Independent Events 86
Problems 91
6 Random Variables 97
6.1 Quantifying the Random Experiment 97
6.2 Cumulative Distribution Function 103
6.3 Functions of a Random Variable 106
6.4 Joint Probability Functions 107
6.5 Marginal Probability Functions 109
6.6 Independence 110
Problems 113
7 Describing Random Variables and their Distributions 121
7.1 Expectation 121
7.2 Laws of Expectation for a Single Random Variable 125
7.3 Variance 126
7.4 Laws of Variance for a Single Random Variable 130
7.5 Standardized Random Variables 131
Problems 132
8 Describing the Joint Behavior of Several Random Variables 139
8.1 Expectation of a Function of Two Random Variables 139
8.2 Covariance 140
8.3 Expectation of the Sum of Several Random Variables 144
8.4 Variance of the Sum of Several Random Variables 145
8.5 Correlation Coefficient 148
8.6 Problems Concerning Several Random Variables 150
8.7 Random Variables Based on Samples 155
Problems 158
9 Special Discrete Probability Models 165
9.1 Binomial Distribution 165
9.2 Waiting Time Distributions 171
9.3 Poisson Distribution 176
9.4 Hypergeometric Distribution 180
9.5 Sums of Binomial Random Variables 186
9.6 Multinomial Distribution 191
Problems 194
10 Statistical Inference 201
10.1 Tests of Hypotheses 201
10.2 Testing for Goodness of Fit 205
10.3 Tests for Comparing Two Groups 209
Problems 211
11 Continuous Distributions 217
11.1 Properties of Continuous Random Variables 217
11.2 Normal Distribution 219
Problems 225
12 Limit Theorems 229
12.1 Chebyshev's Inequality 229
12.2 Law of Large Numbers 232
12.3 Central Limit Theorem 233
12.4 Approximating Discrete Distributions Using the Central Limit Theorem 234
Problems 245
Appendixes
A Summation and Subscripts 249
B Set Theory 253
C Mathematical Induction as a Method of Proof 257
D Binomial Expansions 261
E Infinite Series 267
Tables 271
Answers to Selected Problems 285
Index 315
