Comparisons of variances through the probabilistic mean value theorem and applications

In this paper we adopt the probabilistic mean value theorem in order to study differences of the variances of transformed and stochastically ordered random variables, based on a suitable extension of the equilibrium operator. We also develop a rigorous approach aimed to express the variance of transformed random variables. This is based on a joint distribution which, in turn, involves the variance of the original random variable, as well as its mean residual lifetime and mean inactivity time. Then, we provide applications to the additive hazards model and to some well-known random variables of interest in actuarial science. They deal with a new notion, named `centered mean residual lifetime', and a suitably related stochastic order. Finally, the analysis of the differences of the variances of transformed discrete random variables is also addressed thanks to the use of a discrete version of the equilibrium operator.