Generalized equilibrium distributions

We define the generalized equilibrium distribution, that is the equilibrium distribution of a random variable with support in $\mathbb{R}$. This concept allows us to prove a new probabilistic generalization of the Taylor's theorem. Then, the generalized equilibrium distribution of two ordered random variables is considered and a probabilistic analogue of the mean value theorem is proved. Results regarding distortion-based models and mean-median-mode relations are illustrated as well. Conditions for the unimodality of such distributions are obtained. We show that various stochastic orders and aging classes are preserved through the proposed equilibrium transformations. Further applications are provided in actuarial science, aiming to employ the new unimodal equilibrium distributions for some risks measures, such as Value-at-Risk and Conditional Tail Expectation.