Summary: The paper provides conceptual clarifications for the issues related to the dependence of jointly distributed systems of random entities on external factors. This includes the theory of selective influence as proposed by E. M. Dzhafarov [Selective influence through conditional independence. Psychometrika 68, 7–26 (2003)] and generalized versions of the notions of probabilistic causality [P. Suppes and M. Zanotti, Synthese 48, 191–199 (1981; Zbl 0476.03011)] and dimensionality in the latent variable models [M. V. Levine, J. Math. Psychol. 47, No. 4, 450–466 (2003; Zbl 1077.91042)]. One of the basic observations is that any system of random entities whose joint distribution depends on a factor set can be represented by functions of two arguments: a single factor-independent source of randomness and the factor set itself. In the case of random variables (i.e., real-valued random entities endowed with Borel sigma-algebras), the single source of randomness can be chosen to be any random variable with a continuous distribution (e.g., uniformly distributed between 0 and 1).