Measure-Theoretic Probability

• Verfasst von: Shum, Kenneth [VerfasserIn]
• Titel: Measure-Theoretic Probability
• Titelzusatz: With Applications to Statistics, Finance, and Engineering
• Verf.angabe: by Kenneth Shum
• Ausgabe: 1st ed. 2023.
• Verlagsort: Cham
• Verlagsort: Cham
• Verlag: Springer International Publishing
• Verlag: Imprint: Birkhäuser
• E-Jahr: 2023
• Jahr: 2023.
• Jahr: 2023.
• Umfang: 1 Online-Ressource(XV, 259 p. 33 illus., 25 illus. in color.)
• Gesamttitel/Reihe: Compact Textbooks in Mathematics
• ISBN: 978-3-031-49830-5
• DOI: doi:10.1007/978-3-031-49830-5
• Datenträger: Online-Ressource
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe
• K10plus-PPN: 1881223558
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

Preface -- Beyond discrete and continuous random variables -- Probability spaces -- Lebesgue–Stieltjes measures -- Measurable functions and random variables -- Statistical independence -- Lebesgue integral and mathematical expectation -- Properties of Lebesgue integral and convergence theorems -- Product space and coupling -- Moment generating functions and characteristic functions -- Modes of convergence -- Laws of large numbers -- Techniques from Hilbert space theory -- Conditional expectation -- Levy’s continuity theorem and central limit theorem -- References -- Index.

Abstract

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 collector’s 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.