Power laws and power laws with exponential cut-off are two distinct families of distributions on the positive real half-line. In the present paper, we propose a unified treatment of both families by building a family of distributions that interpolates between them, which we call Interpolating Family (IF) of distributions. Our original construction, which relies on techniques from statistical physics, provides a connection for hitherto unrelated distributions like the Pareto and Weibull distributions, and sheds new light on them. The IF also contains several distributions that are neither of power law nor of power law with exponential cut-off type. We calculate quantile-based properties, moments and modes for the IF. This allows us to review known properties of famous distributions on ${ℝ}^{+}$ and to provide in a single sweep these characteristics for various less known (and new) special cases of our Interpolating Family.

中文翻译：

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从 Pareto 到 Weibull——对 ℝ+ 分布的建设性回顾

幂律和具有指数截断的幂律是正实半线上的两个截然不同的分布族。在本论文中，我们建议通过建立一个在它们之间插值的分布族来统一处理这两个族，我们称之为分布的插值族 (IF)。我们最初的构造依赖于统计物理学的技术，为迄今为止不相关的分布（如帕累托分布和威布尔分布）提供了联系，并为它们提供了新的思路。IF 还包含几个既不是幂律分布也不是具有指数截止类型的幂律分布。我们计算 IF 的基于分位数的属性、矩和模式。这使我们能够回顾著名分布的已知属性${ℝ}^{+}$并在一次扫描中为我们的插值族的各种鲜为人知的（和新的）特例提供这些特征。