A new approach on fractional calculus and probability density function

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 RiemannLiouville fractional integral operators. Besides, we obtain a two-parameter extension of generalized Riemann-Liouville fractional integral inequalities, and provide several modifications in the RiemannLiouville and classical sense. Our ideas and obtained results my stimulate further research in statistical analysis.

A complete description of the distribution of a probability for a given random variable can be obtained by distribution function and density functions. Interestingly, they don't permit us to do comparisons between two distinct distributions. The random variables about mean that particularly portray the appropriation under reasonable conditions is helpful in making comparisons. Knowing the probability function, we can determine the expectation and variance. There are, however, applications wherein the exact forms of probability distributions are not known or are mathematically intractable so that the moments cannot be calculated-as an example, an application in insurance in connection with the insurer's payout on a given contract or group of contracts that follows a mixture or compound probability distribution. It is this problem that motivates researchers to obtain alternative estimations for the expectations and variances of a probability distribution. Applying the mathematical inequalities, some estimations for the expectation and variance of random variables were studied in [47,48].
In 2001, Cerone and Dargomir [49] estimated the bounds of a continuous random variable whose probability density function for the expectation and variance is defined on a finite interval, some integral inequalities have been contemplated for the expectation and variance of a random variable having a probability density function. Kumar [50] derived certain variants for the moments and higher-order moments of a continuous random variable.
The main purpose of the article is to establish some novel estimates for the expectation and variance of the continuous random variables by use of the generalized Riemann-Liouville fractional integral operator, and provide new bounds for certain consequences of the Riemann-Liouville fractional integral, Katugampola fractional integral, conformable fractional integral and Hadamard fractional integral operators by varying the domain as special cases.

Prelude
In this section, we give some basic notions for the generalized Riemann-Liouville fractional integral operators.
Definition 2.1. (See [51]) Let p ≥ 1, r ≥ 0 and υ 1 < υ 2 . Then the function F (ξ) is said to be in If r = 0, then we denote . Then the left-sided and right-sided Riemann-Liouville fractional integrals of order δ > 0 are defined by A generalization of the Riemann-Liouville fractional integrals with respect to another function can be found in [51].
be a finite or infinite real interval, and u(ξ) be an increasing and positive function defined on (η 1 , η 2 ] such that u is continuous on [0, ∞) and u(0) = 0. Then the left-sided and right-sided generalized Riemann-Liouville fractional integrals of a function F with respect to another function u of order δ > 0 are defined by Remark 2.1. From Definition 2.4 we clearly see that (1) If u(ς) = ς, then we get Definition 2.3.
, then it reduces to the conformable fractional integrals operator defined by Jarad et al. [54].
Definition 2.5. Let Y be a random variable with a positive probability density function F defined on [η 1 , η 2 ] and u(ξ) be an increasing and positive function defined on (η 1 , η 2 ]. Then the fractional expectation function E Y,δ (ς) of order δ ≥ 0 is defined by Similarly, we define the fractional expectation function of Y − E(Y) as follows.
Definition 2.6. Let u(ξ) be an increasing and positive function defined on (η 1 , η 2 ]. Then the fractional expectation function E Y−E(Y),δ (ς) of order δ ≥ 0 for a random variable Y − E(Y) with a positive probability density function F defined on [η 1 , η 2 ] is defined by where F : [η 1 , η 2 ] → R + is the probability density function.
If ξ = η 2 , then we present the following definitions.
Definition 2.7. Let η 1 ≥ 0 and u(ξ) be an increasing and positive function defined on (η 1 , η 2 ]. Then the fractional expectation function of order δ ≥ 0 for a random variable Y with a positive probability density function F defined on [η 1 , η 2 ] is defined by Definition 2.8. Let η 1 ≥ 0 and u(ξ) be an increasing and positive function defined on (η 1 , η 2 ]. Then the generalized fractional variance function of order δ ≥ 0 for a random variable Y with a positive probability density function F defined on [η 1 , η 2 ] is defined by ξF (ξ)dξ represents the classical expectation of Y.
If ξ = η 2 , then we have the following definition.
Definition 2.9. Let η 1 ≥ 0 and u(ξ) be an increasing and positive function defined on (η 1 , η 2 ]. Then the generalized fractional variance function of order δ ≥ 0 for a random variable Y with a positive probability density function F : [η 1 , η 2 ] → R + is defined by

Main results
The key aim of this section is to establish several results for the continuous random variable having probability density functions via generalized Riemann-Liouville fractional integral operator. Throughout this paper, we assume that u(ξ) is an increasing and positive function defined defined on [0, ∞) such that u(0) = 0 and u (ξ) is continuous on [0, ∞). Lemma 3.1. Let Y be a continuous random variable with probability density function F : [η 1 , for all δ ≥ 0.
Theorem 3.2 leads to Corollary 3.3 immediately.
Corollary 3.3. Let Y be a continuous random variable with probability density function F : [η 1 , η 2 ] → R + . Then one has the following two conclusion.
Next we provide more general form of Theorem 3.2 by proposing two fractional parameters.
Theorem 3.5. Let Y be a continuous random variable with probability density function F : [η 1 , η 2 ] → R + . Then J u,γ (3.16) Proof. It follows from Theorem 1 of [57] that and (3.18) can be rewritten as Therefore, we get which is the required result.

Conclusions
In the aritcle, we have derived numerous new inequalities in the frame of generalized Riemann-Liouville fractional integral operators via a continuous random variable, our obtained results are the generalizations and refinements of the known results given in [47] and [56]. In the special case of δ = 1, it is worth mentioning that our results can recapture many previously existing operators. Adopting our ideas and approach, researchers can also generate several variants by use of the Hadamard and conformable fractional integral operators and obtain many new inequalities for the probability density functions using different parameters and random variables.