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Power laws and power laws with exponential cut-off are two distinct familiesof distributions on the positive real half-line. In the present paper, wepropose a unified treatment of both families by building a family ofdistributions that interpolates between them, which we call InterpolatingFamily (IF) of distributions. Our original construction, which relies ontechniques from statistical physics, provides a connection for hithertounrelated distributions like the Pareto and Weibull distributions, and shedsnew light on them. The IF also contains several distributions that are neitherof power law nor of power law with exponential cut-off type. We calculatequantile-based properties, moments and modes for the IF. This allows us toreview known properties of famous distributions on $\mathbb{R}^+$ and toprovide in a single sweep these characteristics for various less known (andnew) special cases of our Interpolating Family.

From Pareto to Weibull -- a constructive review of distributions on $\mathbb{R}^+$