Page 9 - Elements of Distribution Theory

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Contents

Preface page xi

1. Properties of Probability Distributions 1
1.1 Introduction 1
1.2 Basic Framework 1
1.3 Random Variables 5
1.4 Distribution Functions 8
1.5 Quantile Functions 15
1.6 Density and Frequency Functions 20
1.7 Integration with Respect to a Distribution Function 26
1.8 Expectation 28
1.9 Exercises 34
1.10 Suggestions for Further Reading 38
2. Conditional Distributions and Expectation 39
2.1 Introduction 39
2.2 Marginal Distributions and Independence 39
2.3 Conditional Distributions 46
2.4 Conditional Expectation 53
2.5 Exchangeability 59
2.6 Martingales 62
2.7 Exercises 64
2.8 Suggestions for Further Reading 67

3. Characteristic Functions 69
3.1 Introduction 69
3.2 Basic Properties 72
3.3 Further Properties of Characteristic Functions 82
3.4 Exercises 90
3.5 Suggestions for Further Reading 93

4. Moments and Cumulants 94
4.1 Introduction 94
4.2 Moments and Central Moments 94
4.3 Laplace Transforms and Moment-Generating Functions 99
4.4 Cumulants 110

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