Continuous Distributional Characterizations via a Single Characterizing Function and Lambert W Conditional Expectation Identities

This paper develops a unified framework for continuous distributional characterizations based on conditional expectations of functions of a random variable. The proposed framework expresses several characterization identities through a single differentiable characterizing function ζ(v), thereby providing necessary and sufficient conditions under which a distribution is uniquely determined by identities of the form Eg(V)|V≤v. Unlike earlier treatments in which related conditional expectation characterizations were handled separately, the present approach places power-, nonlinear power-, exponential-, and logarithmic-type identities under one common formulation. Propositions 1–3 extend earlier conditional expectation characterization results, whereas Proposition 4 provides a related necessary and sufficient characterization with an explicit Lambert W inversion. Propositions 5 and 6 introduce additional necessary and sufficient identities whose explicit inversions involve the Lambert W function. Propositions 2, 5, and 6 are obtained as special cases of a common transformation theorem. A distinctive feature of the proposed results is the systematic use of the Lambert W function to obtain explicit forms of the characterizing function when nonlinear exponential or logarithmic equations arise. The results also clarify cases in which closed-form inversion is difficult and numerical solutions may be required. These characterizations provide useful theoretical tools for identifying continuous probability models and for adapting conditional expectation characterizations to modern generated distribution families. The study fills a gap in the literature by connecting conditional expectation characterizations with modern generated distribution families through a single adaptable function rather than through separate distribution-specific derivations. Analytical applications are provided for several generated families, including TL-G, AGT-G, and OK-G families, illustrating how the proposed propositions can be used to verify and construct distributional characterizations.