Bounded domain distributions naturally come up when random variables should only vary in a finite interval. Some distributions, like beta, occur in a variety of ways, including as order statistics of an underlying uniform distribution or as a model for fractions of some quantity. In general, any unbounded domain distribution can be made to have a bounded domain by using operations such as truncation.
continuous uniform distribution
Bernoulli trial distribution
Distributions can be made to have bounded domains by modifying existing distributions.
truncate any distribution