On lower bounds for the variance of functions of random variables

In this paper, we obtain lower bounds for the variance of a function of random variables in terms of measures of reliability and entropy. Also based on the obtained characterization via the lower bounds for the variance of a function of random variable X, we find a characterization of the weighted function corresponding to density function f(x), in terms of Chernoff-type inequalities. Subsequently, we obtain monotonic relationships between variance residual life and dynamic cumulative residual entropy and between variance past lifetime and dynamic cumulative past entropy. Moreover, we find lower bounds for the variance of functions of weighted random variables with specific weight functions applicable in reliability under suitable conditions.