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A Note on the Moments of Random Variables

A Note on the Moments of Random Variables 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Economic Quality Control de Gruyter

A Note on the Moments of Random Variables

Economic Quality Control , Volume 22 (2) – Oct 1, 2007

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Publisher
de Gruyter
Copyright
Copyright © 2007 by the
ISSN
1869-6147
eISSN
1869-6147
DOI
10.1515/EQC.2007.223
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Economic Quality Controlde Gruyter

Published: Oct 1, 2007

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