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A Note on the Moments of Random Variables

Author

Listed:
  • Sen Surath

    (Stochastikon GmbH, Schießhausstr. 15, D-97072 Würzburg, Germany)

  • von Collani Elart

    (Stochastikon GmbH, Schießhausstr. 15, D-97072 Würzburg, Germany)

Abstract

Any random variable X describing a real phenomenon has necessarily a bounded range of variability implying that the values of the moments determine the probability distribution uniquely. In fact, the range of variability of a random variable restricts the range of the first moment; the value of the first moment limits considerably the range of the second moment; etc. Thus, any knowledge about the values of lower moments may be used without a further sample for drawing inference on the higher moments. In this paper we assume without loss of generality that the range of variability of the random variable X is given by the unit interval. Subsequently, the arising restrictions for the three first moments are derived and the implications with respect to variance and skewness are identified. Moreover, the situation with respect to unimodal random variables is investigated and it is shown that for unimodal probability distributions the third moment yields only marginal additional information.

Suggested Citation

  • Sen Surath & von Collani Elart, 2007. "A Note on the Moments of Random Variables," Stochastics and Quality Control, De Gruyter, vol. 22(2), pages 223-246, January.
    • Handle: RePEc:bpj:ecqcon:v:22:y:2007:i:2:p:223-246:n:7
    DOI: 10.1515/EQC.2007.223
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    References listed on IDEAS

    as
    1. Schelz F. & Sans W. & v. Collani E., 2007. "Improving the Variability Function in Case of a Uni-Modal Probability Distribution," Stochastics and Quality Control, De Gruyter, vol. 22(1), pages 19-39, January.
    2. Sans W. & Zhai X. & Stübner D. & Sen S. & Binder A. & Collani E. v., 2005. "Improving the Variability Function in Case of a Monotonic Probability Distribution," Stochastics and Quality Control, De Gruyter, vol. 20(1), pages 121-142, January.
    3. Stübner Dmitri & Sans Wolfgang & Zhai Xiaomin & Sen Surath & von Collani Elart, 2004. "Measurement Procedures for Improving the Variability Function," Stochastics and Quality Control, De Gruyter, vol. 19(2), pages 215-228, January.
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