Characterization of continuous distributions by properties of conditional variance

In the present paper we study the properties of the left and right truncated variance of a function of a non-negative random variable, that characterize a class of continuous distributions. These properties include characterizations by the relationships the conditional variance has with the truncated expectations and/or the failure rate as well as the lower bound to the conditional variance. It is shown that the characteristic properties are linked to those based on the relationship between the conditional means and the failure rates, discussed in the literature. The lower bound developed here compares favourably with that given by the Cramer–Rao inequality.