Classical intervals have been a very useful tool to analyze uncertain and imprecise models, in spite of operative and interpretative shortcomings. The recent introduction of modal intervals helps to overcome those limitations. In this paper, we apply modal intervals to the field of probability, including properties and axioms that form a theoretical framework applied to the Markovian analysis of Bonus-Malus systems in car insurance. We assume that the number of claims is a Poisson distribution and in order to include uncertainty in the model, the claim frequency is defined as a modal interval; therefore, the transition probabilities are modal interval probabilities. Finally, the model is exemplified through application to two different types of Bonus-Malus systems, and the attainment of uncertain long-run premiums expressed as modal intervals.

中文翻译：

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模态区间概率：在红利系统中的应用

尽管存在操作和解释上的缺陷，但经典区间一直是分析不确定和不精确模型的非常有用的工具。最近引入的模态区间有助于克服这些限制。在本文中，我们将模态区间应用于概率领域，包括属性和公理，这些属性和公理构成了应用于汽车保险中红利-马鲁斯系统的马尔可夫分析的理论框架。我们假设索赔的数量是泊松分布，为了在模型中包含不确定性，索赔频率被定义为模态区间；因此，转移概率是模态区间概率。最后，通过对两种不同类型的 Bonus-Malus 系统的应用以及以模态间隔表示的不确定长期保费的实现来举例说明该模型。