A natural introduction to probability theory
Edition: Second
Publisher: Birkhauser
Published: 2008
Place of publication: Basel CH
Language: eng
Format: softcover, pdf
Copyright date: 2008
Table of contents: title: .; title: Contents; title: Preface to the First Edition viii; title: Preface to the Second Edition x; title: 1 Experiments 1; title: 1.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . 1; title: 1.2 Counting and Combinatorics . . . . . . . . . . . . . . . . . . . . . 6; title: 1.3 Properties of Probability Measures . . . . . . . . . . . . . . . . . . 10; title: 1.4 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . . 12; title: 1.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18; title: 1.6 A First Law of Large Numbers . . . . . . . . . . . . . . . . . . . . 26; title: 1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28; title: 2 Random Variables and Random Vectors 35; title: 2.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 35; title: 2.2 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41; title: 2.3 Expectation and Variance . . . . . . . . . . . . . . . . . . . . . . . 43; title: 2.4 Random Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52; title: 2.5 Conditional Distributions and Expectations . . . . . . . . . . . . . 57; title: 2.6 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 62; title: 2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64; title: 3 Random Walk 71; title: 3.1 Random Walk and Counting . . . . . . . . . . . . . . . . . . . . . 71; title: 3.2 The Arc-Sine Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 75; title: 3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78; title: 4 Limit Theorems 81; title: 4.1 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . 81; title: 4.2 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 85; title: 4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88; title: vi Contents; title: I Intermezzo 89; title: I.1 Uncountable Sample Spaces . . . . . . . . . . . . . . . . . . . . . . 89; title: I.2 An Event Without a Probability?! . . . . . . . . . . . . . . . . . . 90; title: I.3 Random Variables on Uncountable Sample Spaces . . . . . . . . . 92; title: 5 Continuous Random Variables and Vectors 93; title: 5.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93; title: 5.2 Properties of Probability Measures . . . . . . . . . . . . . . . . . . 98; title: 5.3 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . 100; title: 5.4 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105; title: 5.5 Random Vectors and Independence . . . . . . . . . . . . . . . . . . 109; title: 5.6 Functions of Random Variables and Vectors . . . . . . . . . . . . . 113; title: 5.7 Sums of Random Variables . . . . . . . . . . . . . . . . . . . . . . 117; title: 5.8 More About the Expectation; Variance . . . . . . . . . . . . . . . . 118; title: 5.9 Random Variables Which are Neither Discrete Nor Continuous . . 122; title: 5.10 Conditional Distributions and Expectations . . . . . . . . . . . . . 124; title: 5.11 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . 131; title: 5.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131; title: 6 Infinitely Many Repetitions 137; title: 6.1 Infinitely Many Coin Flips and Random Points in (0, 1] . . . . . . 138; title: 6.2 A More General Approach to Infinitely Many Repetitions . . . . . 140; title: 6.3 The Strong Law of Large Numbers . . . . . . . . . . . . . . . . . . 142; title: 6.4 Random Walk Revisited . . . . . . . . . . . . . . . . . . . . . . . . 146; title: 6.5 Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 147; title: 6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151; title: 7 The Poisson Process 153; title: 7.1 Building a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153; title: 7.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158; title: 7.3 The Waiting Time Paradox . . . . . . . . . . . . . . . . . . . . . . 162; title: 7.4 The Strong Law of Large Numbers . . . . . . . . . . . . . . . . . . 164; title: 7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165; title: 8 Limit Theorems 167; title: 8.1 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 167; title: 8.2 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . 169; title: 8.3 Expansion of the Characteristic Function . . . . . . . . . . . . . . 173; title: 8.4 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . 176; title: 8.5 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 179; title: 8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180; title: 9 Extending the Probabilities 183; title: 9.1 General Probability Measures . . . . . . . . . . . . . . . . . . . . . 183; title: Contents vii; title: A Interpreting Probabilities 187; title: B Further Reading 191; title: C Answers to Selected Exercises 193; title: Index 195; title: .