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Generalized equilibrium distributions
- Part of
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- Journal:
- Probability in the Engineering and Informational Sciences , First View
- Published online by Cambridge University Press:
- 31 July 2025, pp. 1-27
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- Article
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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 Taylor’s theorem. Then, the generalized equilibrium distribution of two ordered random variables is considered and a probabilistic analog 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 risk measures, such as Value-at-Risk and Conditional Tail Expectation.
