| Description |
1 online resource (xv, 259 p.) : illustrations (chiefly color) |
| Series |
Compact Textbooks in Mathematics Series |
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Compact textbooks in mathematics
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| Contents |
Intro -- Preface -- Contents -- Notation -- 1 Beyond Discrete and Continuous Random Variables -- 1.1 Discrete and Continuous Random Variables -- 1.2 Random Variables of Mixed Type and Singular Type -- 1.3 Riemann-Stieltjes Integrals -- Problems -- 2 Probability Spaces -- 2.1 Countable Sets -- 2.2 Algebra of Events -- 2.3 Measure Functions -- 2.4 Borel Sets -- 2.5 Vitali Set -- Problems -- 3 Lebesgue-Stieltjes Measures -- 3.1 Pre-measure -- 3.2 Stieltjes Measure Function -- 3.3 Lebesgue-Stieltjes Measures -- 3.4 Null Sets and Complete Measures -- 3.5 Uniqueness of Measure Extension -- Problems |
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4 Measurable Functions and Random Variables -- 4.1 Measurable Functions -- 4.2 Composition of Measurable Functions -- 4.3 Operations with Measurable Functions -- 4.4 Complex-Valued Random Variables -- Problems -- 5 Statistical Independence -- 5.1 Independence of Two Random Variables -- 5.2 Independent Random Variables of Discrete Type or Continuous Type -- 5.3 Independence of More Than Two Random Variables -- 5.4 Borel-Cantelli Lemmas -- 5.5 A Model for a Sequence of Independent Random Variables -- Problems -- 6 Lebesgue Integral and Mathematical Expectation -- 6.1 Simple Functions |
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6.2 Lebesgue Integral of Nonnegative Functions -- 6.3 Lebesgue Integral of Real-Valued and Complex-Valued Functions -- 6.4 Mathematical Expectation of Random Variable -- 6.5 Application: Hat Problem and Ball-and-Bin Model -- Problems -- 7 Properties of Lebesgue Integral and Convergence Theorems -- 7.1 Almost-Everywhere Equality -- 7.2 Fatou's Lemma and Dominated Convergence Theorem -- 7.3 Application: Evaluation of Lebesgue-Stieltjes Integrals -- 7.4 Push-Forward Measure and Change-of-Variable Formula -- 7.5 Expectation of the Product of Two Independent Random Variables -- Problems |
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8 Product Space and Coupling -- 8.1 Coupling -- 8.2 Product Measure and Fubini Theorem -- 8.3 Application: Monge Problem and Kantorovich Problem -- 8.4 Application: Total Variation Distance -- Problems -- 9 Moment Generating Functions and Characteristic Functions -- 9.1 Moments and Moment Generating Functions -- 9.2 Characteristic Functions -- 9.2.1 Properties of Characteristic Functions -- 9.2.2 Inversion Formula -- 9.2.3 Computing Moments from Characteristic Function -- Problems -- 10 Modes of Convergence -- 10.1 Convergence Almost Surely and Convergence in Probability |
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10.2 Convergence in the Mean -- 10.3 Convergence in Distribution and in Total Variation -- 10.4 Convergence of Random Vectors -- 10.5 Application: Continuous Mapping Theorem -- Problems -- 11 Laws of Large Numbers -- 11.1 Some Useful Bounds and Inequalities -- 11.2 Weak Law of Large Numbers -- 11.3 Application: Monte Carlo Integration -- 11.4 Application: Data Compression -- 11.5 Strong Law of Large Numbers -- Problems -- 12 Techniques from Hilbert Space Theory -- 12.1 L2-Norm and Inner Product Space -- 12.2 Closed Subspace and Projection -- 12.3 Orthogonality Principle |
| Summary |
This textbook offers an approachable introduction to measure-theoretic probability, illustrating core concepts with examples from statistics and engineering. The author presents complex concepts in a succinct manner, making otherwise intimidating material approachable to undergraduates who are not necessarily studying mathematics as their major. Throughout, readers will learn how probability serves as the language in a variety of exciting fields. Specific applications covered include the coupon collectors problem, Monte Carlo integration in finance, data compression in information theory, and more. Measure-Theoretic Probability is ideal for a one-semester course and will best suit undergraduates studying statistics, data science, financial engineering, and economics who want to understand and apply more advanced ideas from probability to their disciplines. As a concise and rigorous introduction to measure-theoretic probability, it is also suitable for self-study. Prerequisites include a basic knowledge of probability and elementary concepts from real analysis |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Description based upon print version of record |
|
Description based on online resource; title from digital title page (viewed on March 12, 2024) |
| Subject |
Probabilities.
|
|
probability.
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| Form |
Electronic book
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| ISBN |
9783031498305 |
|
3031498305 |
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