ABSTRACT
To set the stage for the studies in this book, Chapter 1 opens with a short review of the key results on distribution functions from Books I and II. Here the focus is on the connections between distribution functions of random variables and random vectors, and distribution functions induced by Borel measures on ℝ and ℝn. A complete functional characterization of distribution functions on ℝ is then derived, providing a natural link between general probability theory and the discrete and continuous theories commonly encountered. This leads to an investigation into the existence of density functions associated with various distribution functions. Here the Riemann, Lebesgue, and Riemann-Stieltjes integration theories from Book III are recalled to frame this investigation, and the general results to be seen in Book VI using the Book V integration theory are introduced. The chapter ends with a catalog of many common distribution and density functions from the discrete and continuous probability theories, presented within the general frameworks of these theories.
