  For nonnegative random variables with finite means we introduce an analogous
of the equilibrium residual-lifetime distribution based on the quantile
function. This allows to construct new distributions with support (0,1), and to
obtain a new quantile-based version of the probabilistic generalization of
Taylor's theorem. Similarly, for pairs of stochastically ordered random
variables we come to a new quantile-based form of the probabilistic mean value
theorem. The latter involves a distribution that generalizes the Lorenz curve.
We investigate the special case of proportional quantile functions and apply
the given results to various models based on classes of distributions and
measures of risk theory. Motivated by some stochastic comparisons, we also
introduce the `expected reversed proportional shortfall order', and a new
characterization of random lifetimes involving the reversed hazard rate
function.
