  This is the first of two papers that introduce a deformation theoretic
framework to explain and broaden a link between homotopy algebra and
probability theory. In this paper, cumulants are proved to coincide with
morphisms of homotopy algebras. The sequel paper outlines how the framework
presented here can assist in the development of homotopy probability theory,
allowing the principles of derived mathematics to participate in classical and
noncommutative probability theory.
