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Demystifying the Integrated Tail Probability Expectation Formula

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  • Ambrose Lo

Abstract

Calculating the expected values of different types of random variables is a central topic in mathematical statistics. Targeted toward students and instructors in both introductory probability and statistics courses and graduate-level measure-theoretic probability courses, this pedagogical note casts light on a general expectation formula stated in terms of distribution and survival functions of random variables and discusses its educational merits. Often consigned to an end-of-chapter exercise in mathematical statistics textbooks with minimal discussion and presented under superfluous technical assumptions, this unconventional expectation formula provides an invaluable opportunity for students to appreciate the geometric meaning of expectations, which is overlooked in most undergraduate and graduate curricula, and serves as an efficient tool for the calculation of expected values that could be much more laborious by traditional means. For students’ benefit, this formula deserves a thorough in-class treatment in conjunction with the teaching of expectations. Besides clarifying some commonly held misconceptions and showing the pedagogical value of the expectation formula, this note offers guidance for instructors on teaching the formula taking the background of the target student group into account.

Suggested Citation

  • Ambrose Lo, 2019. "Demystifying the Integrated Tail Probability Expectation Formula," The American Statistician, Taylor & Francis Journals, vol. 73(4), pages 367-374, October.
    • Handle: RePEc:taf:amstat:v:73:y:2019:i:4:p:367-374
    DOI: 10.1080/00031305.2018.1497541
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    Cited by:

    1. Carlos Alós-Ferrer & Michele Garagnani, 2022. "Who likes it more? Using response times to elicit group preferences in surveys," ECON - Working Papers 422, Department of Economics - University of Zurich.
    2. Lucas Coffman & Clayton R. Featherstone & Judd B. Kessler, 2024. "A Model of Information Nudges," Boston College Working Papers in Economics 1077, Boston College Department of Economics.
    3. Chen, Le-Yu & Lee, Sokbae, 2023. "Sparse quantile regression," Journal of Econometrics, Elsevier, vol. 235(2), pages 2195-2217.
    4. Albrecher, Hansjörg & Cheung, Eric C.K. & Liu, Haibo & Woo, Jae-Kyung, 2022. "A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 96-118.
    5. Liu, Yang, 2020. "A general treatment of alternative expectation formulae," Statistics & Probability Letters, Elsevier, vol. 166(C).

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