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In this paper, we introduce some new concepts to the field of probability theory: k,s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( k,s\right) $$\end{document}-Riemann–Liouville fractional expectation and variance functions. Some generalized integral inequalities are established for k,s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( k,s\right) $$\end{document}-Riemann–Liouville expectation and variance functions.

be seen in the papers [1][2][3][4]6,9,12,18]. Very recently, new concepts on fractional random variables have been introduced by Dahmani [7]. In this study, Dahmani establishes several integral inequalities for the fractional expectation and the fractional variance functions of a continuous random variable.
First, the Riemann-Liouville fractional integral of order α ≥ 0 for a continuous function f on [a, b] is defined by This integral is motivated by the well-known Cauchy formula: x a dt 1 t 1 a dt 2 . . .
The second is the Hadamard fractional integral introduced by Hadamard [10]. It is given by: The Hadamard integral is based on the generalization of the integral In [13], Katugampola gave a new fractional integration which generalizes both the Riemann-Liouville and Hadamard fractional integrals into a single form. This generalization is based on the observation that, for which gives the following fractional version where α and s = −1 are real numbers. In [8], Diaz and Pariguan have defined new functions called k-gamma and k-beta functions and the Pochhammer k-symbol that is, respectively, generalization of the classical gamma and beta functions and the classical Pochhammer symbol: where (x) n,k is the Pochhammer k-symbol for factorial function. It has been shown that the Mellin transform of the exponential function e − t k k is the k-gamma function, explicitly given by Later, under the above definitions, in [17], Mubeen and Habibullah have introduced k-fractional integral of Riemann-Liouville as follows: Note that when k = 1 in the above integral, then it reduces to the classical Riemann-Liouville fractional integral.

Preliminaries and definitions
In a recent paper [20], Sarikaya et al. have introduced a new fractional integration which generalizes both the k-Riemann-Liouville and Katugampola's fractional integrals into a single form as the following. where Also, in [20], the following results can be seen.

Theorem 2.3 Let f be continuous on
where k denotes the k-gamma function.
Next two theorems, Grüss type inequalities for (k, s)-Riemann-Liouville fractional integrals have been obtained by Set et al. in [21].

Theorem 2.5 Let f and g be two integrable function on
In the following section, we introduce some new concepts for the Riemann-Liouville (k, s)-fractional integral. Also, we give new integral inequalities for the (k, s)-fractional expectation and variance functions of a continuous random variable X having the probability density function f .

(k, s)-Riemann-Liouville fractional integral inequalities
Now, we give the following new definitions for (k, s)-fractional integral operators: The (k, s)-fractional expectation function of order α ≥ 0, for a random variable X with a positive probability density function f defined on [a, b] is defined as Likewise, we will define the (k, s)-fractional expectation function of X − E(X ):

Definition 3.3
The (k, s)-fractional expectation of order α > 0 for a random variable X with a positive probability density function f defined on [a, b] is defined as With a similar logic, we introduce (k, s)-fractional variance function and variance as follows: The (k, s)-fractional variance function of order α ≥ 0 for a random variable X with a positive probability density function f defined on [a, b] is defined as α > 0 and k > 0, s ∈ R\ {−1}.

Definition 3.5
The (k, s) fractional variance of order α ≥ 0, for a random variable X with a positive probability density function f defined on [a, b] is defined as Let us give the following important properties: (i) If we take s = 0 and k = 1 in Definitions 3.1 and 3.4, we obtain the functions of fractional expectation and variance in [7], respectively. (ii) If we take s = 0 and k = 1 in Definitions 3.3 and 3.5, we obtain Definitions 2.4 and 2.6 in [7], respectively.
In this paper, we give new integral inequalities for the (k, s)-fractional expectation and variance functions of a continuous random variable X having the probability density function ( p.d. f.) f .
The first main result is the following theorem:

Theorem 3.6 Let X be a continuous random variable having a probability density function f defined on [a, b]. Then, we have
where Proof Let us define the following identity Taking a function p : [a, b] → R + , multiplying (9) by and integrating the resulting identity with respect to τ from a to t, we can state that Now, multiplying (10) by (s+1 ∈ (a, t) and integrating the resulting identity with respect to ρ from a to t , we can state that In (11), On the other hand, we have Thanks to (12) and (13), we obtain Part (i) of Theorem 3.6.
For Part (ii) of this theorem, we have Then, by (12) and (14), we get the desired inequality (8).
Also we give next theorem: Proof Using the identity (9) in the proof of Theorem 3.6, it follows that Then, for all t ∈ (a, b), choosing (17), we get We have also Thanks to (18) and (19), we obtain (i).
We obtain By (18) and (20), we get the desired inequality (16) which completes the proof.
We also give the following result for Riemann-Liouville (k, s)-fractional integrals: Proof Thanks to (5), we can state that: In (22), if we take p(t) = f (t), g(t) = t − E(X ), t ∈ [a, b] and = b − E (X ), ϕ = a − E (X ), we can state the above inequality as follows: This implies that So, the theorem is proved.
Finally, the next inequality is hold.