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.