Directly Specifying the Power Semicircle Distribution

TL;DR

This paper provides a new proof for a conjecture about the power semicircle distribution, directly specifying its distribution without using the Stieltjes transform, thus clarifying and advancing the theoretical understanding of this distribution.

Contribution

It offers a novel, direct proof of the power semicircle distribution, differing from previous proofs that relied on the Stieltjes transform, and clarifies its distribution.

Findings

Directly specifies the power semicircle distribution.

Provides a proof that does not use the Stieltjes transform.

Clarifies the distribution's properties.

Abstract

A new proof for a newly proved conjecture of Soltani and Roozegar (2012) is provided; our proof does not make any use of the Stieltjes transform unlike the proof of Roozegar and Soltani (2013), and the distribution of power semicircle has been directly specified, contrary to the authors' claim in (Roozegar and Soltani 2013).