Contents

Preface ix

Chapter 1. Some Background and Preliminaries 1

§1.1. The Language of Probability Theory 2

1.1.1. Sample Spaces and Events 3

1.1.2. Probability Measures 4

Exercises for § 1.1 6

§1.2. Finite and Countable Sample Spaces 7

1.2.1. Probability Theory on a Countable Space 7

1.2.2. Uniform Probabilities and Coin Tossing 10

1.2.3. Tournaments 13

1.2.4. Symmetric Random Walk 15

1.2.5. De Moivre’s Central Limit Theorem 17

1.2.6. Independent Events 20

1.2.7. The Arc Sine Law 24

1.2.8. Conditional Probability 27

Exercises for § 1.2 29

§1.3. Some Non-Uniform Probability Measures 32

1.3.1. Random Variables and Their Distributions 32

1.3.2. Biased Coins 33

1.3.3. Recurrence and Transience of Random Walks 36

Exercises for § 1.3 39

§1.4. Expectation Values 40

1.4.1. Some Elementary Examples 45

1.4.2. Independence and Moment Generating Functions 47

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