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In the realm of probability and statistics, the research for accurate and flexible cumulative distribution functions of distributions with support on the unit interval (0, 1) is a prominent challenge. Such distributions are ideal for analyzing proportional or rate-type data, which are ubiquitous in various fields, including biology, finance, environmental science, and reliability analysis. In this article, we introduce a novel collection of eight original cumulative distribution functions that harness the concept of variable-power parametric functions to address this long-standing issue. They may depend on one or several tuning parameters, as well as power, logarithmic, or power-logarithmic functions. Based on them, the collection is enriched by some extended or modified versions with the use of standard transformation schemes (power, type II, transmuted, Topp-Leone, odd Fréchet, etc.). Some graphics illustrate the validity of the main findings. We also discuss how new families of distributions can be generated from the proposed cumulative distribution functions. Overall, this article offers a versatile set of tools for modelling and analyzing (0, 1)-supported data and much more with the generated families. 

cumulative distribution functions, variable-power functions, uniform distribution, family of distributions.