Page 19 - Probability and Statistical Inference

xiv Contents

2.3.4 The Gamma Distribution 84
2.4 Determination of a Distribution via MGF 86
2.5 The Probability Generating Function 88
2.6 Exercises and Complements 89

3 Multivariate Random Variables 99
3.1 Introduction 99
3.2 Discrete Distributions 100
3.2.1 The Joint, Marginal and Conditional Distributions 101
3.2.2 The Multinomial Distribution 103
3.3 Continuous Distributions 107
3.3.1 The Joint, Marginal and Conditional Distributions 107
3.3.2 Three and Higher Dimensions 115
3.4 Covariances and Correlation Coefficients 119
3.4.1 The Multinomial Case 124
3.5 Independence of Random Variables 125
3.6 The Bivariate Normal Distribution 131
3.7 Correlation Coefficient and Independence 139
3.8 The Exponential Family of Distributions 141
3.8.1 One-parameter Situation 141
3.8.2 Multi-parameter Situation 144
3.9 Some Standard Probability Inequalities 145
3.9.1 Markov and Bernstein-Chernoff Inequalities 145
3.9.2 Tchebysheff’s Inequality 148
3.9.3 Cauchy-Schwarz and Covariance Inequalities 149
3.9.4 Jensen’s and Lyapunov’s Inequalities 152
3.9.5 Hölder’s Inequality 156
3.9.6 Bonferroni Inequality 157
3.9.7 Central Absolute Moment Inequality 158
3.10 Exercises and Complements 159

4 Functions of Random Variables and Sampling
Distribution 177
4.1 Introduction 177
4.2 Using Distribution Functions 179
4.2.1 Discrete Cases 179
4.2.2 Continuous Cases 181
4.2.3 The Order Statistics 182
4.2.4 The Convolution 185
4.2.5 The Sampling Distribution 187
4.3 Using the Moment Generating Function 190
4.4 A General Approach with Transformations 192
4.4.1 Several Variable Situations 195
4.5 Special Sampling Distributions 206