A new approach on fractional calculus and probability density function

Shu-Bo Chen; Saima Rashid; Muhammad Aslam Noor; Rehana Ashraf; Yu-Ming Chu; Shu-Bo Chen; Saima Rashid; Muhammad Aslam Noor; Rehana Ashraf; Yu-Ming Chu

doi:10.3934/math.2020451

AIMS Mathematics

2020, Volume 5, Issue 6: 7041-7054. doi: 10.3934/math.2020451

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Research article

A new approach on fractional calculus and probability density function

1.
School of Science, Hunan City University, Yiyang 413000, China
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, COMSATS Universty Islamabad, Isalamabad, Pakistan
4.
Department of Mathematics, Lahore College for Women Universty, Jhangh Campus, Lahore, Pakistan
5.
Department of Mathematics, Huzhou University, Huzhou 313000, China
6.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China

Received: 30 June 2020 Accepted: 30 August 2020 Published: 09 September 2020
MSC : 26A51, 26D10, 26A33

In statistical analysis, oftentimes a probability density function is used to describe the relationship between certain unknown parameters and measurements taken to learn about them. As soon as there is more than enough data collected to determine a unique solution for the parameters, an estimation technique needs to be applied such as "fractional calculus", for instance, which turns out to be optimal under a wide range of criteria. In this context, we aim to present some novel estimates based on the expectation and variance of a continuous random variable by employing generalized Riemann-Liouville fractional integral operators. Besides, we obtain a two-parameter extension of generalized Riemann-Liouville fractional integral inequalities, and provide several modifications in the Riemann-Liouville and classical sense. Our ideas and obtained results my stimulate further research in statistical analysis.
- generalized Riemann-Liouville fractional integral operator,
- integral inequality,
- expectation,
- variance
Citation: Shu-Bo Chen, Saima Rashid, Muhammad Aslam Noor, Rehana Ashraf, Yu-Ming Chu. A new approach on fractional calculus and probability density function[J]. AIMS Mathematics, 2020, 5(6): 7041-7054. doi: 10.3934/math.2020451

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Abstract

In statistical analysis, oftentimes a probability density function is used to describe the relationship between certain unknown parameters and measurements taken to learn about them. As soon as there is more than enough data collected to determine a unique solution for the parameters, an estimation technique needs to be applied such as "fractional calculus", for instance, which turns out to be optimal under a wide range of criteria. In this context, we aim to present some novel estimates based on the expectation and variance of a continuous random variable by employing generalized Riemann-Liouville fractional integral operators. Besides, we obtain a two-parameter extension of generalized Riemann-Liouville fractional integral inequalities, and provide several modifications in the Riemann-Liouville and classical sense. Our ideas and obtained results my stimulate further research in statistical analysis.

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