A Novel Extension of Randomly Weighted Average

TL;DR

This paper introduces a new extension for the problem of a uniform random variable between two independent variables, identifying new classes of distributions via generalized Stieltjes transforms and analyzing their properties using Schwartz distribution theory.

Contribution

It presents a novel extension of the randomly weighted average problem and characterizes new distribution classes through advanced transform techniques.

Findings

Identified new classes of randomly weighted average distributions.

Applied generalized Stieltjes transforms to characterize distributions.

Used Schwartz distribution theory to analyze properties of the new distributions.

Abstract

We study a well-known problem concerning a random variable $Z$ uniformly distributed between two independent random variables. A new extension has been introduced for this problem and fairly large classes of randomly weighted average distributions are identified by their generalized Stieltjes transforms. In this article we employ the Schwartz distribution theory for finding distributions of this extension; we also study some of their properties.