We introduce a notion of the derivative with respect to a function, not necessarily related
to a probability distribution, which generalizes the concept of derivative as proposed by
Lebesgue [14]. The differential calculus required to solve linear differential equations using
this notion of the derivative is included in the paper. The definition given here may also be
considered as the inverse operator of a modified notion of the Riemann–Stieltjes integral.
Both this unified approach and the results of differential calculus allow us to characterize
distributions in terms of three different types of conditional expectations. In applying these
results, a test of goodness of fit is also indicated. Finally, two characterizations of a general
Poisson process are included. Specifically, a useful result for the homogeneous Poisson
process is generalized.