Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications

Weerawat Sudsutad,Nantapat Jarasthitikulchai,Jutarat Kongson,Chatthai Thaiprayoon,Jehad Alzabut

doi:10.3390/math10040573

Abstract

This study investigates a variety of novel estimations involving the expectation, variance, and moment functions of continuous random variables by applying a generalized proportional fractional integral operator. Additionally, a continuous random variable with a probability density function is presented in context of the proportional Riemann–Liouville fractional integral operator. We establish some interesting results of the proportional fractional expectation, variance, and moment functions. In addition, constructive examples are provided to support our conclusions. Meanwhile, we discuss a few specific examples that may be extrapolated from our primary results.

Highlights

Integral inequality is the motivating force behind the modern mathematical analysis perspective
Several researchers have recently investigated a variety of fractional applications for a continuous random variable (C.R.V) with a probability density function (P.D.F)
The generalized proportional fractional integral operators, definitions, and introductory facts are introduced which will be used throughout this work

Summary

Introduction

Integral inequality is the motivating force behind the modern mathematical analysis perspective It has been used in a variety of fields, including probability theory and statistical problems, mathematics, physics, and applied sciences; see [1,2]. Fractional calculus deals with fractional-order (non-integer-order) differential and integral operators, which establish phenomenon modeling that is increasingly realistic in real-world problems It has properly specified the term “memory”, in mathematics, physics, chemistry, biology, mechanics, electricity, finance, economics, and control theory. Several researchers have recently investigated a variety of fractional applications for a continuous random variable (C.R.V) with a probability density function (P.D.F). Dahmani [34] applied the RL-fractional integral operator to analyze integral inequality results for the fractional expectation (F.E) and fractional variance (F.V) functions of the C.R.V in 2014. The conclusion of this paper is presented in the last section

Preliminaries

Main Results

Some Examples

Conclusions