Certain Family of Some Beta Distributions Arising from Distribution of Randomly Weighted Average

TL;DR

This paper derives the exact distribution of weighted averages of independent beta variables using a novel integral transform, expanding understanding of their probabilistic properties and providing new analytical tools.

Contribution

Introduces a new integral transformation similar to the generalized Stieltjes transform to derive the exact distribution of weighted beta averages.

Findings

Derived the exact distribution for weighted averages of beta variables.

Developed a new integral transform with various properties.

Applied the method to several examples of the new family.

Abstract

We give the exact distribution of the average of n independent beta random variables weighted by the selected cuts of (0, 1) by the order statistics of a random sample of size n-1 from the uniform distribution U(0,1), for each n. A new integral transformation that is similar to generalized Stieltjes transform is given with various properties. The result of Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach. Statist Probab Lett. 2012, 82(5):1012-1020] with this new transform and also integral representation of the Gauss-hypergeometric function in some parts are employed to achieve the exact distribution. Several examples of the new family are investigated.