  We present a novel, yet rather simple construction within the traditional
framework of Scott domains to provide semantics to probabilistic programming,
thus obtaining a solution to a long-standing open problem in this area. Unlike
current main approaches that employ some probability measures or continuous
valuations on non-standard or rather complex structures, we use the Scott
domain of random variables from a standard sample space -- the unit interval or
the Cantor space -- to any given Scott domain. The map taking any such random
variable to its corresponding probability distribution provides an effectively
given, Scott continuous surjection onto the probabilistic power domain of the
underlying Scott domain, establishing a new basic result in classical domain
theory. We obtain a Cartesian closed category by enriching the category of
Scott domains to capture the equivalence of random variables on these domains.
The construction of the domain of random variables on this enriched category
forms a strong commutative monad, which is suitable for defining the semantics
of probabilistic programming.
