Demystifying the Integrated Tail Probability Expectation Formula

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.

KEYWORDS:

• Covariance
• Expected value
• Fubini's theorem
• Hoeffding's formula
• Integration by parts

Additional information

Funding

This work was supported by a research fund provided by the Department of Statistics and Actuarial Science, The University of Iowa, and a Centers of Actuarial Excellence (CAE) Research Grant (2018-2021) from the Society of Actuaries (SOA). Any opinions, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the SOA.