  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 $\mathbb{R}^+$ and to
provide in a single sweep these characteristics for various less known (and
new) special cases of our Interpolating Family.
