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A Remark on the Alternative Expectation Formula

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  • Liang Hong

Abstract

Students in their first course in probability will often see the expectation formula for nonnegative continuous random variables in terms of the survival function. The traditional approach for deriving this formula (using double integrals) is well-received by students. Some students tend to approach this using integration by parts, but often get stuck. Most standard textbooks do not elaborate on this alternative approach. We present a rigorous derivation here. We hope that students and instructors of the first course in probability will find this short note helpful.

Suggested Citation

  • Liang Hong, 2012. "A Remark on the Alternative Expectation Formula," The American Statistician, Taylor & Francis Journals, vol. 66(4), pages 232-233, November.
    • Handle: RePEc:taf:amstat:v:66:y:2012:i:4:p:232-233
    DOI: 10.1080/00031305.2012.726934
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    Cited by:

    1. Piyush Kant Rai & Anu Sirohi, 2020. "Alternative approach to moments of order statistics from one-parameter Weibull distribution," Statistics in Transition New Series, Polish Statistical Association, vol. 21(1), pages 169-178, March.
    2. Ogasawara, Haruhiko, 2018. "The inverse survival function for multivariate distributions and its application to the product moment," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 71-76.
    3. Song, Pingfan & Tan, Changchun & Wang, Shaochen, 2019. "On the moment generating function for random vectors via inverse survival function," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 345-350.
    4. Liu, Yang, 2020. "A general treatment of alternative expectation formulae," Statistics & Probability Letters, Elsevier, vol. 166(C).
    5. Joel E. Cohen, 2015. "Markov's Inequality and Chebyshev's Inequality for Tail Probabilities: A Sharper Image," The American Statistician, Taylor & Francis Journals, vol. 69(1), pages 5-7, February.
    6. Rai Piyush Kant & Sirohi Anu, 2020. "Alternative approach to moments of order statistics from one-parameter Weibull distribution," Statistics in Transition New Series, Statistics Poland, vol. 21(1), pages 155-162, March.

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