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Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

Author

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  • Subhabrata Chakraborti
  • Felipe Jardim
  • Eugenio Epprecht

Abstract

Undergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.

Suggested Citation

  • Subhabrata Chakraborti & Felipe Jardim & Eugenio Epprecht, 2019. "Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula," The American Statistician, Taylor & Francis Journals, vol. 73(2), pages 191-194, April.
    • Handle: RePEc:taf:amstat:v:73:y:2019:i:2:p:191-194
    DOI: 10.1080/00031305.2017.1356374
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    Cited by:

    1. Alessandro Barbiero & Asmerilda Hitaj, 2024. "Discrete half-logistic distributions with applications in reliability and risk analysis," Annals of Operations Research, Springer, vol. 340(1), pages 27-57, September.
    2. Liu, Yang, 2020. "A general treatment of alternative expectation formulae," Statistics & Probability Letters, Elsevier, vol. 166(C).
    3. Hong-Jiang Wu & Ying-Ying Zhang & Han-Yu Li, 2023. "Expectation identities from integration by parts for univariate continuous random variables with applications to high-order moments," Statistical Papers, Springer, vol. 64(2), pages 477-496, April.

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