Fractional Probability Theory of Arbitrary Order

Abstract

A generalization of probability theory is proposed by using the Riemann–Liouville fractional integrals and the Caputo and Riemann–Liouville fractional derivatives of arbitrary (non-integer and integer) orders. The definition of the fractional probability density function (fractional PDF) is proposed. The basic properties of the fractional PDF are proven. The definition of the fractional cumulative distribution function (fractional CDF) is also suggested, and the basic properties of these functions are also proven. It is proven that the proposed fractional cumulative distribution functions generate unique probability spaces that are interpreted as spaces of a fractional probability theory of arbitrary order. Various examples of the distributions of the fractional probability of arbitrary order, which are defined on finite intervals of the real line, are suggested.

1. Introduction

Fractional calculus can be considered as a generalization of the standard calculus of differential and integral operators of an integer order to the case of an arbitrary order [1,2,3,4,5,6,7]. Integrals and derivatives of arbitrary orders, which are described by positive parameters, satisfy the fundamental theorems of fractional calculus [8]. Differential and integral operators of arbitrary order have been known for more than three hundred years [9,10,11,12,13,14,15]. Equations with fractional derivatives and integrals of non-integer orders are applied to describe various physical processes in nonlinear dynamics, statistical physics, quantum mechanics, and other fields (see the books [16,17,18,19,20,21,22,23,24,25] and the handbooks [26,27]). For the correct description of probabilistic processes in various fields of physics, and first of all in fractional statistical mechanics and nonlocal statistical mechanics, the generalizations of probability theory to the nonlocal case [28] can play an important role.

The most-important motivation for constructing a fractional generalization of probability theory is the development of fractional (nonlocal) statistical mechanics. This is based on the fact that a rigorous fractional (nonlocal) probability theory is necessary for a self-consistent construction of fractional (nonlocal) statistical mechanics. This is primarily connected with the fact that the basic equation of statistical mechanics, which is called the Liouville equation, describes the law of conservation and the variation of the probability density. The first attempts to construct elements of the fractional probability theory and its applications in statistical mechanics were made in papers since 2004 (for example, see [16,29,30,31,32,33] and the references therein). In statistical mechanics, nonlocal processes with the power-law nonlocality are considered by using integral and differential equations with derivatives of non-integer order (see, for example, [16,32,33,34,35,36]). To obtain these basic equations such as the Liouville equation, the Bogoliubov equation, the Fokker–Planck equation, the conservation of probability is applied in phase space [32,33]. The fractional Liouville equations with differential operators of non-integer orders with respect to coordinates and momenta were suggested in [16,32,33,34,36]. For the first time, nonlocal (fractional) generalizations of the concepts of the average value and nonlocal generalizations of the normalization conditions for probability density functions were suggested in 2004 [16,29,30,31]. The fractional and nonlocal probability theories can be important for nonlocal statistical mechanics [37,38,39,40,41], fractional kinetics and anomalous transport [21,22,42,43,44,45], and non-Markovian quantum dynamics [46,47].

A nonlocal generalization of probability theory based on the use of the general fractional calculus has recently been proposed. However, some important cases were outside the scope of the proposed probability theory. In the article [28], a generalization of the theory of probability to the nonlocal case was proposed by using the Luchko form of the general fractional calculus [48,49] (see also [50,51,52,53,54,55,56]). The applications of the proposed nonlocal probability theory to nonlocal statistical mechanics were suggested in [36]. However, the general fractional calculus allows us to formulate a generalization of probability theory only to continuous distributions on the semiaxis $[0,\infty )$. To construct a probability theory for the distribution on finite intervals $[a,b]$ of the real line $\mathbb{R}=(-\infty ,+\infty )$, one should construct an extension of the general fractional calculus to this case. Now, this problem remains open and awaits its solution (see Section 2.4 “Open problems” in [57], p. 3255). Due to the absence of such an extension, in this paper, it is proposed to use the fractional calculus of the Riemann–Liouville fractional integrals and derivatives. Note that using the generalized Taylor series in the Trujillo–Rivero–Bonilla form [58], the wide class of operator kernels can be represented as a series of power-law kernels. Therefore, equations with Riemann–Liouville fractional integrals and derivatives can be considered as an approximation of the equations with general operator kernels [59] for a wide class of the operator kernels.

In this work, not only the fractional theory of probability at the finite intervals of the real line is proposed, but also a generalization of the probability theory to arbitrary orders. This is achieved through the use of fractional integrals and fractional derivatives of arbitrary orders $\alpha >0$. In this paper, the Riemann–Liouville fractional integrals and the Riemann–Liouville and Caputo fractional derivatives, which form a fractional calculus [1,2,3,4,5], are used.

In Section 2, fractional integrals and derivatives of arbitrary orders are described. Some important properties and a set of functions are described. The fundamental theorems of fractional calculus are given. The description of these elements of fractional calculus, which are used in fractional probability theory, is based on the books [1,4]. In Section 3.1, the characteristic properties of the cumulative distribution function are discussed by using some theorems from the books [60,61]. In Section 3.2, the fractional probability density function is defined and its properties are considered in detail. In Section 3.3, fractional cumulative distribution functions are defined and their properties are considered in detail. Using the properties of the fractional cumulative distribution function, two theorems, which are basic for the existence of the fractional probability of arbitrary order $\alpha >0$, are suggested. In Section 4, examples of distributions of fractional probability on finite intervals $[a,b]$ are proposed. In Section 5, fractional average (mean) values are defined and examples of the calculations are offered. Section 6 gives a brief conclusion.

2. Preliminaries

In this section, we give the main definitions and formulate some basic theorems that are used in the construction of a fractional probability theory of arbitrary order.

2.1. Riemann–Liouville Fractional Integrals

In this subsection, the definitions of sets of functions and the fractional integration are presented.

Definition 1. 

Let $[a,b]$, where $-\infty \le a<b\le \infty$, be a finite or infinite interval of the real axis $\mathbb{R}=(-\infty ,\infty )$.

Then, $L_p(a,b)$, where $1\le p<\infty$, is the set of those Lebesgue measurable functions on $[a,b]$, for which

$${\parallel f\parallel }_{L_p(a,b)}={\left({\int }_a^b{\left|f\left(u\right)\right|}^{p}du\right)}^{1/p}<\infty .$$

(1)

Let us define fractional integral of order $\alpha >0$ by the following definition (see Definition 2.1 of [1], p. 33).

Definition 2. 

Let $f\left(x\right)\in L_1(a,b)$. Then, the integral:

$$\left(I_{a+}^{\alpha }f\right)\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_a^x{(x-u)}^{\alpha -1}f\left(u\right)du,$$

(2)

where $\alpha >0$, $x>a$, is called the Riemann–Liouville fractional integral of order α.

The fractional integral (2) is defined for functions $f\left(x\right)\in L_1(a,b)$, existing almost everywhere ([1], p. 34).

Theorem 2.6 of [1], pp. 48–51, describes the mapping properties of the fractional integral operator (2) in the spaces $L_p(a,b)$ of summable functions. This theorem states that operators of the Riemann–Liouville fractional integration form a semigroup in $L_p(a,b)$, $p\ge 1$, which is continuous in the uniform topology for all $\alpha >0$ and strongly continuous for all $\alpha \ge 0$.

Theorem 1. 

Note that the Riemann–Liouville fractional integral operators (2) are bounded in the spaces $L_p(a,b)$ with $p\ge 1$. The following inequality:

$$\parallel \left(I_{a+}^{\alpha }f\right)\left(x\right){\parallel }_{L_p(a,b)}\le \frac{{(b-a)}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}{\parallel f\left(x\right)\parallel }_{L_p(a,b)}<\infty$$

(3)

holds for $\alpha >0$, where $1\le p<\infty$.

Inequality (3) was proven in [1] as estimation Equation 2.72 (see the proof of Theorem 2.6 in [1], p. 48).

Theorem 1 means that the Riemann–Liouville fractional integrals keep the space $L_p(a,b)$ invariant.

The fractional integral $I_{a+}^{\alpha }\phi$ can be “better” than the function $\phi \in L_p(a,b)$. It is proven that the fractional integral belongs to $L_q(a,b)$ with $q>p$, if $0<\alpha <1/p$ (see [1], p. 66).

Theorem 2. 

If $0<\alpha <1$, $1<p<{\alpha }^{-1}$, then the fractional integration operator $I_{a+}^{\alpha }$ is bounded from $L_p(a,b)$ into $L_q(a,b)$ with $q=p/(1-\alpha p)$.

Theorem 2 was proven in [1], pp. 66–67, as Theorem 3.5.

2.2. Riemann–Liouville Fractional Derivatives

Let $[a,b]$ be a finite interval, and let $AC\left(\right[a,b\left]\right)$ be the space of absolutely continuous functions on $[a,b]$.

The space $AC\left(\right[a,b\left]\right)$ coincides with the space of primitives of Lebesgue summable functions: A function $f\left(x\right)$ belongs to the set $AC\left(\right[a,b\left]\right)$, if and only if there exists a function $\phi \left(x\right)$ such that

$$f\left(x\right)=c+{\int }_a^x\phi \left(u\right)du,\mathrm{where}{\int }_a^b\left|\phi \left(u\right)\right|du<\infty .$$

(4)

Therefore, absolutely continuous functions have a summable derivative $f^{\left(1\right)}\left(x\right)$ almost everywhere. Note that absolute continuity does not follow from the existence of a summable derivative almost everywhere.

Fractional differentiation of the order $\alpha$ is defined (see Definition 2.2 of [1], p. 35) in the form as follows:

Definition 3. 

Let $f\left(x\right)\in AC\left(\right[a,b\left]\right)$.

For the functions $f\left(x\right)$ given on the interval $[a,b]$, the expression:

$$\left(D_{a+}^{\alpha }f\right)\left(x\right)=\frac{1}{\mathrm{\Gamma }(1-\alpha )}\frac{d}{dx}{\int }_a^x{(x-u)}^{-\alpha }f\left(u\right)du,$$

(5)

is called the Riemann–Liouville fractional derivative of order $\alpha \in (0,1)$.

Theorem 3. 

Let $f\left(x\right)\in AC\left(\right[a,b\left]\right)$. Then, the Riemann–Liouville fractional derivative (5) exists almost everywhere for $0<\alpha <1$.

Moreover,

$$\left(D_{a+}^{\alpha }f\right)\left(x\right)\in L_q(a,b),1\le q<{\alpha }^{-1}$$

(6)

and

$$\left(D_{a+}^{\alpha }f\right)\left(x\right)=\frac{1}{\mathrm{\Gamma }(1-\alpha )}{\int }_a^x{(x-u)}^{-\alpha }{f}^{\left(1\right)}\left(u\right)du+\frac{{(x-a)}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}f\left(a\right),$$

(7)

where $f^{\left(1\right)}\left(u\right)=df\left(u\right)/du$.

Theorem 3 was proven in [1], pp. 35–36, as Lemma 2.2.

Definition 4. 

Let a function $f\left(x\right)$ have continuous derivatives of integer orders up to the order $n\in \mathbb{N}$ on interval $[a,b]$ with $f^{(n-1)}\left(x\right)\in AC\left([a,b]\right)$.

The set of such functions is denoted as $A{C}^n\left([a,b]\right)$.

The fractional derivatives of orders $\alpha \ge 1$ are defined ([4], p. 70) in the following way.

Definition 5. 

Let $f\left(x\right)\in A{C}^n\left([a,b]\right)$.

Then, the expression:

$$\left(D_{a+}^{\alpha }f\right)\left(x\right)={\left(\frac{d}{dx}\right)}^{n-1}\left(D_{a+}^{\alpha -n+1}f\right)\left(x\right)={\left(\frac{d}{dx}\right)}^n\left(I_{a+}^{n-\alpha }f\right)\left(x\right),$$

(8)

is called the Riemann–Liouville fractional derivative of order α, where $n=\left[\alpha \right]+1$; the symbol $\left[\alpha \right]$ means an integer part of number α, and $\left\{\alpha \right\}$ is a non-integer part of α, $0\le \left\{\alpha \right\}<1$.

Theorem 4. 

The space $A{C}^n\left([a,b]\right)$ consists of those and only those functions $f\left(x\right)$, which can be given in the form:

$$f\left(x\right)=\sum _{k=0}^{n-1}{c}_k{(x-a)}^k+\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_a^x{(x-u)}^{n-1}\phi \left(u\right)du,$$

(9)

where $\phi \left(x\right)\in L_1(a,b)$ and

$$c_k=\frac{f^{\left(k\right)}\left(a\right)}{\mathrm{\Gamma }(n+1)},$$

(10)

where $n\in \mathbb{R}$.

Theorem 4 was proven in [1], p. 39, as Lemma 2.4 (see also Lemma 1.1 in [4], p. 2).

If a finite measure is used (for example, a probability measure) and $1\le p\le q\le \infty$, then $L_q(a,b)subset{L}_p(a.b)$. For example, the condition $\phi \left(x\right)\in L_q(a,b)$ can be used instead of $\phi \left(x\right)\in L_1(a,b)$.

Note that

$$\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_a^x{(x-u)}^{n-1}\phi \left(u\right)du=\left(I_{a+}^n\phi \right)\left(x\right),$$

(11)

where $n\in \mathbb{R}$.

Theorem 5. 

Let $f\left(x\right)\in A{C}^n\left([a,b]\right)$, $\alpha \ge 0$ and $n=\left[\alpha \right]+1$.

Then, the fractional derivative $\left(D_{a+}^{\alpha }f\right)\left(x\right)$ exists almost everywhere and can be given in the form:

$$\left(D_{a+}^{\alpha }f\right)\left(x\right)=\sum _{k=0}^{n-1}\frac{f^{\left(k\right)}(a+)}{\mathrm{\Gamma }(k+1-\alpha )}{(x-a)}^{k-\alpha }+\frac{1}{\mathrm{\Gamma }(n-k)}{\int }_a^x{(x-u)}^{n-\alpha -1}{f}^{\left(n\right)}\left(u\right)du.$$

(12)

The proof of Theorem 5 was given in [1], pp. 39–40, as Theorem 2.2.

Remark 1. 

In the standard theory of probability, the probability density function (PDF) has the dimension that is inverse to the dimension of the random variable, i.e., $\left[f_{\alpha =1}\right]=\left[x\right]$. The probability and standard cumulative distribution function (CDF) are dimensionless. For the correct fractional generalization of standard probability theory, the physical dimensions of the integrals and derivatives of non-integer orders should be specified [28]. To have the standard dimension of the fractional PDF and fractional CDF in fractional probability theory, the dimensions of the FD and FI or the order $\alpha \in (n-1,n)$ should coincide with the dimensions of the derivative and integral of the integer order $n\in \mathbb{N}$, respectively. Then, the kernels of the fractional integrals should be dimensionless and the dimensions of the kernels of the fractional derivatives should be equal to ${\left[x\right]}^{-n}$. As a result, the kernels should be replaced in the following way:

$$\frac{x^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}⟶\frac{{\left(\lambda x\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)},\frac{x^{n-\alpha -1}}{\mathrm{\Gamma }(n-\alpha )}⟶\frac{\lambda {\left(\lambda x\right)}^{n-\alpha -1}}{\mathrm{\Gamma }(n-\alpha )},$$

(13)

where $\lambda >0$ and $\left[\lambda \right]={\left[x\right]}^{-1}$.

In this paper, to simplify the consideration, the parameter λ will not be used in the equations, assuming it equal to unity. If necessary, this parameter is easy to restore in the equations.

2.3. Fundamental Theorems of Fractional Calculus

Fractional integrals (2) and fractional derivatives (8) form a fractional calculus. The relationship of these fractional operators is described by the first and second fundamental theorems of the fractional calculus.

Definition 6. 

Let $\alpha >0$. A function $f\left(x\right)\in L_1(a,b)$ is said to have a summable fractional derivative $\left(D_{a+}^{\alpha }f\right)\left(x\right)$, if $\left(I_{a+}^{n-\alpha }f\right)\left(x\right)\in A{C}^n\left([a,b]\right)$, $n=\left[\alpha \right]+1$

Let us define the space $I_{a+}^{\alpha }\left(L_q\right)$ (see Definition 2.3 of [1], p. 43).

Definition 7. 

Let a function $f\left(x\right)$ be represented by the fractional integral of order α of a summable function $\phi \left(x\right)$:

$$f\left(x\right)=\left(I_{a+}^{\alpha }\phi \right)\left(x\right),$$

(14)

where $\phi \left(x\right)\in L_q(a,b)$, $1\le p<\infty$.

The set of such functions is denoted as $I_{a+}^{\alpha }\left(L_q\right)$.

The characterization of the space $I_{a+}^{\alpha }\left(L_p\right)$ is given by the following theorem generalizing Theorem 2.1 ([1], p. 43).

Theorem 6. 

In order that $f\left(x\right)\in I_{a+}^{\alpha }\left(L_q\right)$ with $1\le q\le \infty$ and $\alpha >0$, it is necessary and sufficient that

$$\left(I_{a+}^{n-\alpha }f\right)\left(x\right)\in A{C}^n\left([a,b]\right),\underset{x\to a+}{lim}\frac{d^k}{d{x}^k}\left(I_{a+}^{n-\alpha }f\right)\left(x\right)=0$$

(15)

for $k=0,1,\dots ,n-1$.

Theorem 6 was proven in [1], p. 43, as Theorem 2.3, where $f\left(x\right)\in I_{a+}^{\alpha }\left(L_1\right)$ is considered. In Theorem 6, the property $L_q\subset L_1$ is used for $q>1$.

Theorem 7. 

(First fundamental theorem of fractional calculus)

Let $f\left(x\right)$ be a summable function, $f\left(x\right)\in L_p(a,b)$ with $1\le p\le \infty$ and $\alpha >0$.

Then, the equality:

$$\left(D_{a+}^{\alpha }{I}_{a+}^{\alpha }f\right)\left(x\right)=f\left(x\right)$$

(16)

holds almost everywhere on $[a,b]$.

Theorem 7 was proven in [1], pp. 44–45, as Theorem 2.4. See also Lemma 2.4 in [4], p. 74.

Theorem 8. 

(Second fundamental theorem of fractional calculus)

Let $f\left(x\right)\in L_1(a,b)$ have a summable derivative $\left(D_{a+}^{\alpha }f\right)\left(x\right)$, i.e., $\left(I_{a+}^{n-\alpha }f\right)\left(x\right)\in A{C}^n\left([a,b]\right)$.

Then, the equality:

$$\left(I_{a+}^{\alpha }{D}_{a+}^{\alpha }f\right)\left(x\right)=f\left(x\right)-\sum _{k=1}^n\frac{{(x-a)}^{\alpha -k}}{\mathrm{\Gamma }(\alpha -k+1)}\left(\frac{d^{n-k}}{d{x}^{n-k}}{I}_{a+}^{n-\alpha }f\right)(a+).$$

(17)

holds almost everywhere on $[a,b]$.

If $f\left(x\right)\in I_{a+}^{\alpha }\left(L_p\right)$, with $1\le p\le \infty$ and $\alpha >0$, then the equality

$$\left(I_{a+}^{\alpha }{D}_{a+}^{\alpha }f\right)\left(x\right)=f\left(x\right)$$

(18)

is satisfied for all $x\in [a,b]$.

Theorem 8 was proven in [1], pp. 44–45, as Theorem 2.4 with $f\left(x\right)\in I_{a+}^{\alpha }\left(L_1\right)$ for Equation (18). See also Lemma 2.5 in [4], pp. 74–75, where $f\left(x\right)\in I_{a+}^{\alpha }\left(L_p\right)$ for Equation (18).

2.4. Caputo Fractional Derivatives and Fundamental Theorems

In fractional calculus, in addition to the Riemann–Liouville fractional derivatives, the Caputo fractional derivatives are actively used.

Definition 8. 

Let $\alpha \ge 0$, and let $f^{\left(n\right)}\left(x\right)\in L_1(a,b)$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the operator:

$$\left(D_{a+}^{\alpha ,*}f\right)\left(x\right)={\int }_a^x\frac{{(x-u)}^{n-\alpha -1}}{\mathrm{\Gamma }(n-\alpha )}{f}^{\left(n\right)}\left(u\right)du=\left(I_{a+1}^{n-\alpha }{D}^{n}f\right)\left(x\right),$$

(19)

is called the Caputo fractional derivative of order α, where $D^{n}f\left(x\right)=f^{\left(n\right)}\left(x\right)=d^{n}f\left(x\right)/d{x}^n$ is the derivative of the integer order $n\in \mathbb{N}$.

The Caputo fractional derivative and the Riemann–Liouville fractional derivative are related by the following relation (see [4], p. 91) at the intersection of their domains of definition:

$$(D_{a+}^{\alpha ,*}f)\left(x\right)=(D_{a+}^{\alpha }f)\left(x\right)-\sum ^{n-1}\frac{{(x-a)}^{k-\alpha }}{\mathrm{\Gamma }(k-\alpha +1)}{f}^{\left(k\right)}\left(a\right),$$

(20)

where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

For non-integer values of $\alpha >0$, the Caputo fractional derivative coincides with the Riemann–Liouville fractional derivatives in the case $f^{\left(k\right)}\left(a\right)=0$ for $k=0,1,\dots ,n-1$.

Theorem 9. 

Let $\alpha \ge 0$, and let $f\left(x\right)\in A{C}^n\left([a,b]\right)$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the Caputo fractional derivative $\left(D_{a+}^{\alpha ,*}f\right)\left(x\right)$ exist almost everywhere on $[a,b]$.

Theorem 9 was proven in [4], pp. 92–93, as Theorem 2.1, and in [5], p. 53, as Lemma 3.4.

Theorem 10. 

(First fundamental theorem of fractional calculus with the Caputo FD)

Let $\alpha >0$, and let $f\left(x\right)\in C[a,b]$ or $f\left(x\right)\in L_{\infty }(a,b)$.

Then, the equality:

$$\left(D_{a+}^{\alpha ,*}{I}_{a+}^{\alpha }f\right)\left(x\right)=f\left(x\right)$$

(21)

is satisfied for all $x\in (a,b)$.

Theorem 10 was proven in [4], pp. 95–96, as Lemma 2.21, and in [5], p. 53, as Theorem 3.7.

Theorem 11. 

(Second fundamental theorem of fractional calculus with the Caputo FD)

Let $\alpha >0$, and let $f\left(x\right)\in A{C}^n\left([a,b]\right)$ or $f\left(x\right)\in C^n[a,b]$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the equality:

$$\left(I_{a+}^{\alpha }{D}_{a+}^{\alpha ,*}f\right)\left(x\right)=f\left(x\right)-\sum _{k=0}^{n-1}\frac{{(x-a)}^k}{\mathrm{\Gamma }(k+1)}{f}^{\left(k\right)}\left(a\right)$$

(22)

is satisfied for all $x\in (a,b)$.

Theorem 11 was proven in [4], pp. 96–97, as Lemma 2.22, and in [5], p. 54, as Theorem 3.8.

Using Equation (20) and Theorem 11, the following statement can be proven.

Corollary 1. 

Let $\alpha >0$, and let $f\left(x\right)\in A{C}^n\left([a,b]\right)$ or $f\left(x\right)\in C^n[a,b]$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Let the following condition be satisfied:

$$f^{\left(k\right)}\left(a\right)=0\mathit{for}k=0,1,\dots ,n-1.$$

(23)

Then, the equalities:

$$\left(I_{a+}^{\alpha }{D}_{a+}^{\alpha ,*}f\right)\left(x\right)=f\left(x\right),$$

(24)

$$\left(D_{a+}^{\alpha ,*}f\right)\left(x\right)=\left(D_{a+}^{\alpha }f\right)\left(x\right)\alpha \notin \mathbb{N}$$

(25)

are satisfied for all $x\in (a,b)$.

3. From Fractional Probability Density to Fractional Probability Space

3.1. Preliminaries: Characteristic Properties of Cumulative Distribution Function

For the construction of a fractional probability theory of an arbitrary order, theorems that allow one to construct a probability space based on the characteristic properties of the cumulative distribution function are important. In this subsection, these properties and theorems are described.

Consider the probability (measurable) space $(\mathbb{R},\mathcal{B}(\mathbb{R}),P)$ on the real line $\mathbb{R}$, where $\mathcal{B}\left(\mathbb{R}\right)$ is a system of Borel sets on $\mathbb{R}$ and P is a probability measure [60,61].

Definition 9. 

Let $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$ be the real line with the system $\mathcal{B}\left(\mathbb{R}\right)$ of Borel sets. Let $P=P\left(A\right)$ be a probability measure defined on the Borel subset A of the real line $\mathbb{R}$. If $A=(-\infty ,x]$, where $x\in \mathbb{R}$, then

$$F\left(x\right)=P(-\infty ,x]$$

(26)

is called the cumulative distribution function.

The following theorem describes the characteristic properties of the cumulative distribution function.

Theorem 12. 

Let $[a,b]$, where $-\infty <a<b<+\infty$, be a finite interval of the real line $\mathbb{R}$, and let $F\left(x\right)$ be a cumulative distribution function of a random variable X.

Then, $F\left(x\right)$ satisfies the following properties:

I. 

Right-continuity: $F\left(x\right)$ is a continuous function on the right and has a limit on the left at each $x\in \mathbb{R}$.

$$\underset{x\to x_0+}{lim}F\left(x\right)=F\left(x_0\right).$$

(27)

II. 

Monotonicity: $F\left(x\right)$ is a non-decreasing function. If $a\le x_1<x_2\le b$, then $F\left(x_1\right)\le F\left(x_2\right)$.

III. 

The behavior at interval boundaries:

$$\underset{x\to a+}{lim}F\left(x\right)=0,$$

(28)

$$\underset{x\to b-}{lim}F\left(x\right)=1.$$

(29)

Theorem 12 was proven in [60], p. 185, and in the book [61], p. 34.

One can formulate a theorem inverse to Theorem 12. This theorem shows what properties a function must have in order to be a cumulative distribution function.

Theorem 13. 

Let $F=F\left(x\right)$ be a function on the real line $\mathbb{R}$, which satisfies Conditions I, II, and III.

Then, there exists a unique probability (measurable) space $(\mathbb{R},\mathcal{B}(\mathbb{R}),P)$ and a random variable X such that

$$P(X\le x)=F\left(x\right),$$

(30)

$$P(x_1<X\le x_2)=P(x_1,x_2]=F\left(x_2\right)-F\left(x_1\right)$$

(31)

for all $x_1$, $x_2$ such that $-\infty \le x_1<x_2<\infty$.

Theorem 13 was proven in the book [61], p. 35, as Theorem 3.2.1, and it was also described in [60], p. 185, as Theorem 1.

As a result, one can state that any function $F\left(x\right)$ on the real line $\mathbb{R}$ that satisfies Conditions I, II, and III is a cumulative distribution function.

Theorem 13 allows us to consider the following special case.

Corollary 2. 

Let $[a,b]$, where $-\infty \le a<b\le \infty$, be a finite or infinite interval of the real axis $\mathbb{R}=(-\infty ,\infty )$.

Let $F=F\left(x\right)$ be a function on $[a,b]\subset \mathbb{R}$, which satisfies the following conditions:

N1. $F\left(x\right)$ belongs to the space $A{C}^n\left([a,b]\right)$.

N2. Monotonicity: $F\left(x\right)$ is non-decreasing function:

$$\frac{d}{dx}F\left(x\right)\ge 0.$$

(32)

N3a. The behavior on the left boundary of the interval:

$$\underset{x\to a+}{lim}F\left(x\right)=0.$$

(33)

N3b. The behavior on the right boundary of the interval:

$$\underset{x\to b-}{lim}F\left(x\right)=1.$$

(34)

Then, the function $F=F\left(x\right)$ satisfies Conditions I, II, and III.

Note that a function that is absolutely continuous on an interval is uniformly continuous and, therefore, continuous. The reverse is not true.

3.2. Fractional Probability Density Function

Let us define a set of non-negative functions.

Definition 10. 

Let a function $f\left(x\right)$ be represented by the fractional integral of order $n-\alpha$ of a summable non-negative function $\phi \left(x\right)$ in the form:

$$f\left(x\right)=\left(I_{a+}^{n-\alpha }\phi \right)\left(x\right),$$

(35)

where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$:

$$\phi \left(x\right)\in L_q(a,b),\mathrm{with}1\le q<\infty ,$$

(36)

and

$$\phi \left(x\right)\ge 0,\mathit{for}\mathit{all}x\in [a,b]$$

(37)

The set of such functions is denoted as $I_{a+}^{n-\alpha ,+}\left(L_q\right)$.

Let us describe the properties of the functions $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$.

Property 1. 

Let $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the function $f\left(x\right)$ is bounded in the spaces $L_q(a,b)$ with $1\le q<\infty$:

$${\parallel f\left(x\right)\parallel }_{L_q(a,b)}=\frac{{(b-a)}^{n-\alpha }}{\mathrm{\Gamma }(n-\alpha +1)}{\parallel \phi \left(x\right)\parallel }_{L_q(a,b)}<\infty$$

(38)

Proof. 

Using that $\phi \left(x\right)\in L_q(a,b)$ and Theorem 1, one can obtain:

$${\parallel f\left(x\right)\parallel }_{L_q(a,b)}=\parallel \left(I_{a+}^{n-\alpha }\phi \right)\left(x\right){\parallel }_{L_q(a,b)}\le \frac{{(b-a)}^{n-\alpha }}{\mathrm{\Gamma }(n-\alpha +1)}{\parallel \phi \left(x\right)\parallel }_{L_q(a,b)}<\infty ,$$

(39)

where $\alpha >0$ and $q\ge 1$. □

Property 2. 

Let $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $1<q<{(n-\alpha )}^{-1}$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, $f\left(x\right)\in L_p(a,b)$, and the function $f\left(x\right)$ is bounded in the spaces $L_p(a,b)$ with $p=q/(1-(n-\alpha \left)q\right)$:

$$f\left(x\right)\in L_p(a,b)\mathit{with}p=\frac{q}{1-(n-\alpha )q}.$$

(40)

Proof. 

Theorem 2 (Theorem 3.5 in [1], p. 66) can be reformulated in the following form. If $n-1<\alpha <n$ and $1<q<1/(n-\alpha )$, then the fractional integration operator $I_{a+}^{n-\alpha ,+}$ is bounded from $L_q(a,b)$ into $L_p(a,b)$ with $p=q/(1-(n-\alpha \left)q\right)$. This statement was proven in [1], pp. 66–67, as Theorem 3.5 for $0<n-\alpha <1$. □

Property 3. 

Let $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $1<q<1/(n-\alpha )$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the function $f\left(x\right)$ is non-negative:

$$f\left(x\right)\ge 0,\mathit{for}\mathit{all}x\in [a,b].$$

(41)

Proof. 

If $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$, then $\phi \left(x\right)\ge 0$ for all $x\in [a,b]$, Using that $\phi \left(x\right)\ge 0$, the fact ${(x-u)}^{n-\alpha -1}\ge 0$ that holds, if $x>u$ for all $x\in [a,b]$ and the properties of integration, Equation (35) gives $f\left(x\right)=\left(I_{a+}^{n-\alpha }\phi \right)\left(x\right)\ge 0$ for all $x\in [a,b]$. □

Property 4. 

Let $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $1<q<1/(n-\alpha )$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the Riemann–Liouville fractional integral $\left(I_{a+}^{\alpha }f\right)\left(x\right)$ of the function $f\left(x\right)$ exists almost everywhere, and it is bounded in the spaces $L_p(a,b)$ with $p=q/(1-(n-\alpha \left)q\right)$.

Proof. 

Using Property 2 for $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$, one can see that $f\left(x\right)\in L_p(a,b)$. Then, using Theorem 1, one can state that the Riemann–Liouville fractional integral $\left(I_{a+}^{\alpha }f\right)\left(x\right)$ is bounded in the spaces $L_p(a,b)$ with $p=q/(1-(n-\alpha \left)q\right)\ge 1$, such that

$$\parallel \left(I_{a+}^{\alpha }f\right)\left(x\right){\parallel }_{L_p(a,b)}\le \frac{{(b-a)}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}{\parallel f\left(x\right)\parallel }_{L_p(a,b)}<\infty$$

(42)

 □

Property 1. 

Let $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $1<q<1/(n-\alpha )$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the Riemann–Liouville fractional integral $\left(I_{a+}^{\alpha }f\right)\left(x\right)$ of the function $f\left(x\right)$ is non-negative:

$$\left(I_{a+}^{\alpha }f\right)\left(x\right)\ge 0,\mathit{for}\mathit{all}x\in [a,b].$$

(43)

Proof. 

Using Property 3, which states $f\left(x\right)\ge 0$, the fact ${(x-u)}^{\alpha -1}\ge 0$ that holds if $x>u$ for all $x\in [a,b]$, and the properties of integration, Equation (35) gives $\left(I_{a+}^{\alpha }f\right)\left(x\right)\ge 0$ for all $x\in [a,b]$. □

Property 6. 

Let $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $1<q<1/(n-\alpha )$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the Riemann–Liouville fractional integral $\left(I_{a+}^{\alpha }f\right)\left(x\right)$ of the function $f\left(x\right)$ belongs to the space $A{C}^n\left([a,b]\right)$:

$$\left(I_{a+}^{\alpha }f\right)\left(x\right)\in A{C}^n\left([a,b]\right),$$

(44)

if the condition:

$$\underset{x\to a+}{lim}\frac{d^k}{d{x}^k}\left(I_{a+}^{\alpha }f\right)\left(x\right)=0$$

(45)

is satisfied for $k=0,1,\dots ,n-1$.

Proof. 

The condition $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$ means that $f\left(x\right)$ can be given as

$$f\left(x\right)=\left(I_{a+}^{n-\alpha ,+}\phi \right)\left(x\right).$$

(46)

Then,

$$\left(I_{a+}^{\alpha }f\right)\left(x\right)=\left(I_{a+}^{\alpha }{I}_{a+}^{n-\alpha }\phi \right)\left(x\right).$$

(47)

The semigroup property of the fractional integrals (see Lemma 2.3 in [4], p. 73) in the form:

$$\left(I_{a+}^{\alpha }{I}_{a+}^{\beta }\phi \right)\left(x\right)=\left(I_{a+}^{\alpha +\beta }\phi \right)\left(x\right),$$

(48)

is satisfied at almost every point $x\in [a,b]$ for $\phi \in L_q(a,b)$ with $1\le q\le \infty$. Note that if $\alpha +\beta >1$, then Relation (48) holds at any point of $[a,b]$. Using (48), Equation (47) gives

$$\left(I_{a+}^{\alpha }f\right)\left(x\right)=\left(I_{a+}^n\phi \right)\left(x\right),$$

(49)

where $\phi \left(x\right)\in L_q(a,b)$. Therefore, the fractional integral $\left(I_{a+}^{\alpha }f\right)\left(x\right)$ can be given in the form

$$\left(I_{a+}^{\alpha }f\right)\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(n\right)}{\int }_a^x{(x-u)}^{n-1}\phi \left(u\right)du,$$

(50)

for all $x\in (a,b)$. Then, using Theorem 4, one can obtain that $\left(I_{a+}^{\alpha }f\right)\left(x\right)\in A{C}^n\left([a,b]\right)$, if the equations $\left(I_{a+}^{\alpha }f\right)(a+)=0$ are satisfied for $k=0,1,\dots ,n-1$. □

Definition 11. 

Let a function $f\left(x\right)$ belong to the space $I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $L_q=L_q(a,b)$ and $1<q<1/(n-\alpha )$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the function:

$$f_{\alpha }\left(x\right)=\frac{1}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}f\left(x\right),$$

(51)

where $x\in [a,b]$, is called the fractional probability density function (fractional PDF) of order $\alpha >0$.

Using the fractional probability density function, one can define a fractional cumulative distribution function. Then, using the fractional CDF, one can define a fractional probability of arbitrary order $\alpha >0$.

3.3. Fractional Cumulative Distribution Function

Let us define a function that will be interpreted as a fractional cumulative distribution function.

Definition 12. 

Let a function $f\left(x\right)$ belong to the space $I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $L_q=L_q(a,b)$ and $1<q<1/(n-\alpha )$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the function:

$$F_{\alpha }\left(x\right)=\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(x\right)=\frac{1}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}\left(I_{a+}^{\alpha }f\right)\left(x\right)$$

(52)

is called the fractional cumulative distribution function (fractional CDF) of order $\alpha >0$.

Let us describe the properties of the fractional cumulative distribution functions.

Using Equations (51) and (52), Properties 2, 5, and 6 can be expressed through the fractional cumulative distribution function in the form of Properties 7–9, respectively.

Property 7. 

Let $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $1<q<1/(n-\alpha )$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the fractional CDF $F_{\alpha }\left(x\right)$ belongs to the space $I_{a+}^{\alpha ,+}\left(L_p\right)$,

$$F_{\alpha }\left(x\right)\in I_{a+}^{\alpha ,+}\left(L_p\right),$$

(53)

where

$$p=\frac{q}{1-(n-\alpha )q}.$$

(54)

Property 8. 

Let $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $1<q<1/(n-\alpha )$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the fractional CDF $F_{\alpha }\left(x\right)$ is a non-negative function:

$$F_{\alpha }\left(x\right)\ge 0,\mathit{for}\mathit{all}x\in [a,b].$$

(55)

Property 9. 

Let $f\left(x\right)\in I_{a+}^{n-\alpha ,+}\left(L_q\right)$ with $1<q<1/(n-\alpha )$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the fractional CDF $F_{\alpha }\left(x\right)$ belongs to the space $A{C}^n\left([a,b]\right)$:

$$F_{\alpha }\left(x\right)\in A{C}^n\left([a,b]\right),$$

(56)

if the conditions:

$$F_{\alpha }^{\left(k\right)}(a+)=0$$

(57)

are satisfied for $k=0,1,\dots ,n-1$, where $F_{\alpha }^{\left(k\right)}\left(x\right)=d^k{F}_{\alpha }\left(x\right)/d{x}^k$.

Let us describe some statements following from Properties 7–9.

Property 10. 

Let $F_{\alpha }\left(x\right)\in A{C}^n\left([a,b]\right)$, where $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Then, the Caputo fractional derivative $\left(D_{a+}^{\alpha ,*}F\right)\left(x\right)$ exists almost everywhere on $[a,b]$.

Proof. 

The property follows directly from Theorem 9 and Property 9. □

Property 11. 

Let $F_{\alpha }\left(x\right)\in A{C}^n\left([a,b]\right)$, $\alpha \ge 0$, and $n=\left[\alpha \right]+1$.

Then, the Riemann–Liouville fractional derivative $\left(D_{a+}^{\alpha }{F}_{\alpha }\right)\left(x\right)$ exists almost everywhere and can be given in the form:

$$\left(D_{a+}^{\alpha }{F}_{\alpha }\right)\left(x\right)=\left(D_{a+}^{\alpha ,*}{F}_{\alpha }\right)\left(x\right)+\sum _{k=0}^{n-1}\frac{{(x-a)}^{k-\alpha }}{\mathrm{\Gamma }(k-\alpha +1)}{F}_{\alpha }^{\left(k\right)}(a+).$$

(58)

Proof. 

The property follows directly from Theorem 5 and Property 9. □

Property 12. 

Let $F_{\alpha }\left(x\right)\in A{C}^n\left([a,b]\right)$, $\alpha \ge 0$, and $n=\left[\alpha \right]+1$.

If the the following property is satisfied:

$$F_{\alpha }^{\left(k\right)}(a+)=\underset{x\to a+}{lim}{F}_{\alpha }^{\left(k\right)}\left(x\right)=0,$$

(59)

then the Caputo fractional derivative $\left(D_{a+}^{\alpha ,*}f\right)\left(x\right)$ and the Riemann–Liouville fractional derivative $\left(D_{a+}^{\alpha ,*}f\right)\left(x\right)$ coincide:

$$\left(D_{a+}^{\alpha ,*}{F}_{\alpha }\right)\left(x\right)=\left(D_{a+}^{\alpha }{F}_{\alpha }\right)\left(x\right).$$

(60)

Proof. 

Using Condition (59), Equation (58) gives Equality (60). □

One can state that, in the fractional probability theory, the equation:

$$\underset{x\to a+}{lim}{F}_{\alpha }^{\left(k\right)}\left(x\right)=0,$$

(61)

replaces the standard condition:

$$\underset{x\to a+}{lim}{F}_{\alpha }\left(x\right)=0.$$

(62)

Using the definition of the fractional CDF $F_{\alpha }\left(x\right)$ and Theorem 7, which describes the first fundamental theorem of fractional calculus, one can see the following property.

Property 13. 

Let $F_{\alpha }\left(x\right)\in I_{a+}^{\alpha ,+}\left(L_p(a,b)\right)$ with $1\le p\le \infty$ and $\alpha >0$.

Then, the Riemann–Liouville fractional derivative of the fractional CDF $F_{\alpha }\left(x\right)=\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(x\right)$ is equal to the fractional PDF $f_{\alpha }\left(x\right)$:

$$\left(D_{a+}^{\alpha }{F}_{\alpha }\right)\left(x\right)=f_{\alpha }\left(x\right)$$

(63)

almost everywhere on $[a,b]$.

Proof. 

Using the definition of the fractional CDF $F_{\alpha }\left(x\right)$ and Theorem 7, which describes the first fundamental theorem of fractional calculus, one can obtain Equation (63). □

As a result, the Caputo fractional derivative $(D_{a+}^{\alpha ,*}{F}_{\alpha })\left(x\right)$ and the Riemann–Liouville fractional derivative $(D_{a+}^{\alpha ,*}{F}_{\alpha })\left(x\right)$ of the fractional CDF $F_{\alpha }\left(x\right)$ coincide and are equal to the fractional probability density function:

$$(D_{a+}^{\alpha ,*}{F}_{\alpha })\left(x\right)=(D_{a+}^{\alpha }{F}_{\alpha })\left(x\right)=f_{\alpha }\left(x\right).$$

(64)

Property 14. 

Let $F_{\alpha }\left(x\right)$ be a fractional CDF. Then, the fractional normalization condition can be given in the form:

$$\underset{x\to b-}{lim}{F}_{\alpha }\left(x\right)=1.$$

(65)

Proof. 

Using Equation (52), the fractional normalization condition gives

$$\underset{x\to b-}{lim}{F}_{\alpha }\left(x\right)=\underset{x\to b-}{lim}\frac{1}{F\left(b\right)}F\left(x\right)=1.$$

(66)

 □

The non-decreasing property can be described in the following way.

Property 15. 

Let $F_{\alpha }\left(x\right)\in I_{a+}^{\alpha ,+}\left(L_p(a,b)\right)$ with $1\le p\le \infty$ and $\alpha >0$, and let $f_{\alpha }\left(x\right)\in I_{a+}^{\alpha ,+}\left(L_q(a,b)\right)$.

Then, the non-decreasing property has the form:

$$\frac{d}{dx}{F}_{\alpha }\left(x\right)\ge 0.$$

(67)

for all $x\in [a,b]$.

Proof. 

Using the definition of the fractional CDF $F_{\alpha }\left(x\right)$ and the fact that $f_{\alpha }\left(x\right)\in I_{a+}^{\alpha }\left(L_q\right)$, i.e., $f_{\alpha })\left(x\right)=I_{a+}^{n-\alpha }\phi )\left(x\right)$, the first-order derivative of the fractional CDF is in the form:

$$\frac{d}{dx}{F}_{\alpha }\left(x\right)=\frac{d}{dx}\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(x\right)=\frac{d}{dx}\left(I_{a+}^{\alpha }{I}_{a+}^{n-\alpha }\phi \right)\left(x\right).$$

(68)

Then, using the semi-group property of the fractional integrals, one can obtain

$$\frac{d}{dx}{F}_{\alpha }\left(x\right)=\frac{d}{dx}\left(I_{a+}^n\phi \right)\left(x\right)=\left(I_{a+}^{n-1}\phi \right)\left(x\right),$$

(69)

where $n\in \mathbb{N}$ and $\left(I_{a+}^0\phi \right)\left(x\right)=\phi \left(x\right)$. Using that $\phi \left(x\right)\ge 0$ and the fact ${(x-u)}^{n-1}\ge 0$ for $x>u$, one can obtain

$$\frac{d}{dx}{F}_{\alpha }\left(x\right)\ge 0$$

(70)

for all $x\in [a,b]$. □

Theorem 6 can be reformulated for a fractional cumulative distribution function $F_{\alpha }\left(x\right)=\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(x\right)$. As a result, we obtain an additional property of the fractional CDF.

Property 16. 

In order that $F_{\alpha }\left(x\right)\in I_{a+}^{\alpha ,+}\left(L_p\right)$ with $1\le p\le \infty$ and $\alpha >0$, it is necessary and sufficient that

$$\left(I_{a+}^{n-\alpha }{F}_{\alpha }\right)\left(x\right)\in A{C}^n\left([a,b]\right),\underset{x\to a+}{lim}\frac{d^k}{d{x}^k}\left(I_{a+}^{n-\alpha }{F}_{\alpha }\right)\left(x\right)=0$$

(71)

for $k=0,1,\dots ,n-1$.

Proof. 

Property 16 was proven in [1], p. 43, as Theorem 2.3. □

Property 17. 

If $F_{\alpha }\left(x\right)\in I_{a+}^{\alpha ,+}\left(L_p\right)$, with $1\le p\le \infty$ and $\alpha >0$, then the equality:

$$\left(I_{a+}^{\alpha }{D}_{a+}^{\alpha }{F}_{\alpha }\right)\left(x\right)=\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(x\right)=F_{\alpha }\left(x\right)$$

(72)

is satisfied for all $x\in [a,b]$.

Proof. 

Property (17) is based on Theorem 8. It was proven in [1], pp. 43–44, as Theorem 2.4 and in [4], p. 74, as Lemma 2.4. Theorem 8 states that the equality:

$$\left(I_{a+}^{\alpha }{D}_{a+}^{\alpha }F\right)\left(x\right)=F\left(x\right)$$

(73)

is satisfied for all $x\in [a,b]$, If $F\left(x\right)\in I_{a+}^{\alpha }\left(L_p\right)$, with $1\le p\le \infty$ and $\alpha >0$. □

As a result, the described properties of the fractional cumulative distribution function allow us to formulate two following theorems, which are basic for the existence of the fractional probability of arbitrary order $\alpha >0$.

Theorem 14. 

Let $[a,b]$, where $-\infty \le a<b\le \infty$, be a finite or infinite interval of the real axis $\mathbb{R}=(-\infty ,\infty )$.

Let $F=F_{\alpha }\left(x\right)$ be a function on $[a,b]\subset \mathbb{R}$, which satisfies the following conditions:

F1. $F_{\alpha }\left(x\right)$ belongs to the space $A{C}^n\left([a,b]\right)$.

F2. Monotonicity: $F_{\alpha }\left(x\right)$ is a non-decreasing function:

$$\frac{d}{dx}{F}_{\alpha }\left(x\right)\ge 0\mathit{for}\mathit{all}x\in [a,b]$$

(74)

F3a. The behavior on the left boundary of the interval:

$$\underset{x\to a+}{lim}{F}_{\alpha }^{\left(k\right)}\left(x\right)=0$$

(75)

for $k=0,1,\dots ,n-1$.

F3b. The behavior on the right boundary of the interval:

$$\underset{x\to b-}{lim}{F}_{\alpha }\left(x\right)=1,$$

(76)

Then, the function $F=F_{\alpha }\left(x\right)$ satisfies Conditions I, II, and III.

Note that a function that is absolutely continuous on an interval is uniformly continuous, and therefore continuous. The reverse is not true.

Using Theorems 13 and 14, one can obtain the following theorem.

Theorem 15. 

Let $\mathrm{\Omega }=[a,b]$, where $-\infty \le a<b\le \infty$, be a finite or infinite interval of the real axis $\mathbb{R}=(-\infty ,\infty )$.

Let a function $F_{\alpha }\left(x\right)\in I_{a+}^{\alpha ,+}\left(L_p\left(\mathrm{\Omega }\right)\right)$ with $1\le p\le \infty$ and $\alpha >0$ be defined by the equation:

$$F_{\alpha }\left(x\right)=\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(x\right),$$

(77)

where $f_{\alpha }\left(x\right)$ belongs to the space $I_{a+}^{n-\alpha ,+}\left(L_q(a,b)\right)$ with $n=\left[\alpha \right]+1$ for non-integer values of $\alpha >0$ and $n=\alpha$ for integer values of $\alpha >0$.

Let $F_{\alpha }\left(x\right)$ satisfy the condition:

$$\underset{x\to a+}{lim}{F}_{\alpha }^{\left(k\right)}\left(x\right)=0$$

(78)

for $k=0,1,\dots ,n-1$.

Then, there exists a unique probability space $(\mathrm{\Omega },\mathcal{B}(\mathrm{\Omega }),P)$ and a random variable X such that

$$P(X\le x)=F_{\alpha }\left(x\right),$$

(79)

$$P(x_1<X\le x_2)=P(x_1,x_2]=F_{\alpha }\left(x_2\right)-F_{\alpha }\left(x_1\right)$$

(80)

for all $x_1,x_2\in \mathrm{\Omega }$ such that $a\le x_1<x_2\le b$.

The probability of such a type is called the fractional probability.

Let us note that the fractional probability is defined by the equation:

$$P(x_1,x_2]=F_{\alpha }\left(x_2\right)-F_{\alpha }\left(x_1\right)=\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(x_2\right)-\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(x_1\right)=$$

$${\int }_a^{x_2}\frac{{(x_2-x)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{f}_{\alpha }\left(x\right)dx-{\int }_a^{x_1}\frac{{(x_1-x)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{f}_{\alpha }\left(x\right)dx.$$

(81)

In the general case, this expression (81) cannot be written in the form:

$$P(x_1,x_2]\ne \left(I_{x_1}^{\alpha }{f}_{\alpha }\right)\left(x_2\right)={\int }_{x_1}^{x_2}\frac{{(x_2-x)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{f}_{\alpha }\left(x\right)dx$$

(82)

This follows from the inequality:

$${\int }_a^{x_2}\frac{{(x_2-x)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{f}_{\alpha }\left(x\right)dx-{\int }_a^{x_1}\frac{{(x_1-x)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{f}_{\alpha }\left(x\right)dx\ne {\int }_{x_1}^{x_2}\frac{{(x_2-x)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{f}_{\alpha }\left(x\right)dx.$$

(83)

The identity can only be obtained for the case $\alpha =1$.

As a result of Theorem 15, one can state that any fractional cumulative distribution function $F_{\alpha }\left(x\right)$, which is defined in Theorem 15, generates a unique probability space that describes fractional probability theory of arbitrary order $\alpha >0$.

4. Examples of Fractional Distributions on Interval $[a,b]$

In this section, some examples of fractional probability distributions are considered. As reference information on the Riemann–Liouville fractional integrals of various functions, Table 9.1 of [1], p. 173, is used. In this section, the fractional probability of arbitrary order is considered distributed on the final interval. Let $[a,b]$, where $-\infty <a<b<\infty$, be a finite interval of the real axis $\mathbb{R}=(-\infty ,\infty )$.

4.1. Fractional Distributions of Power-Law and Uniform Types

For the function:

$$f\left(x\right)={(x-a)}^{\beta -1},$$

(84)

where $x>a$ and $\beta >0$, one can obtain (see Equation 1 in Table 9.1 of [1], p. 173) the equation:

$$F\left(x\right)=\left(I_{a+}^{\alpha }f\right)\left(x\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{(x-a)}^{\alpha +\beta -1}.$$

(85)

For $x=b$, Equation (85) has the form:

$$\left(I_{a+}^{\alpha }f\right)\left(b\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{(b-a)}^{\alpha +\beta -1}.$$

(86)

As a result, the fractional distribution of the power-law type is described by the functions as follows.

The fractional PDF is defined as

$$f_{\alpha }\left(x\right)=\frac{1}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}f\left(x\right)=\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }\left(\beta \right)}\frac{{(x-a)}^{\beta -1}}{{(b-a)}^{\alpha +\beta -1}}.$$

(87)

The fractional CDF is defined as

$$F_{\alpha }\left(x\right)={\left(\frac{x-a}{b-a}\right)}^{\alpha +\beta -1},$$

(88)

where $\alpha >0$, $\beta >0$, and $\alpha +\beta -1>0$.

One can see that the equalities:

$$F_{\alpha }\left(a\right)=0,$$

(89)

$$F_{\alpha }\left(b\right)=1$$

(90)

are satisfied, if $\alpha +\beta -1>0$, where $\alpha >0$ and $\beta >0$.

For the case $\beta =1$, the fractional distribution of the power-law type gives the fractional distribution of the uniform type, which is described by the following functions.

The fractional PDF is defined as

$$f_{\alpha }\left(x\right)=\frac{1}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}{f}_{\alpha }\left(x\right)=\frac{\mathrm{\Gamma }(\alpha +1)}{{(b-a)}^{\alpha +\beta -1}}.$$

(91)

The fractional CDF is defined as

$$F_{\alpha }\left(x\right)={\left(\frac{x-a}{b-a}\right)}^{\alpha },$$

(92)

where $\alpha >0$.

Let us give a simple example of calculating the fractional probability for the fractional distribution of the power-law type on the finite interval $[a,b]$. Let $[c,d]\subset [a,b]$ such that $a<c<d<b$. Then,

$$P_{\alpha }[c,d]=F_{\alpha }\left(d\right)-F_{\alpha }\left(c\right)={\left(\frac{d-a}{b-a}\right)}^{\alpha +\beta -1}-{\left(\frac{c-a}{b-a}\right)}^{\alpha +\beta -1}$$

(93)

For the fractional distribution of the uniform type ($\beta =0$), one can obtain

$$P_{\alpha }[c,d]={\left(\frac{d-a}{b-a}\right)}^{\alpha }-{\left(\frac{c-a}{b-a}\right)}^{\alpha }$$

(94)

Note that one can see that Equation (94) cannot be represented as

$$P_{\alpha }[c,d]\ne {\left(\frac{d-c}{b-a}\right)}^{\alpha }.$$

(95)

This illustrates the following inequality:

$$P_{\alpha }[c,d]=\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(d\right)-\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(d\right)\ne \left(I_{c+}^{\alpha }{f}_{\alpha }\right)\left(d\right)$$

(96)

for $\alpha \ne 1$. For the standard uniform distribution ($\alpha =1$), Equation (94) gives the standard expression:

$$P_{\alpha =1}[c,d]=\frac{d-c}{b-a},$$

(97)

where $0<d-c<b-a$.

4.2. Fractional Distribution of Two-Parameter Power-Law Type

For the function:

$$f\left(x\right)={(x-a)}^{\beta -1}{(b-x)}^{\gamma -1},$$

(98)

one can obtain (see Equation 3 in Table 9.1 of [1], p. 173) the equation:

$$F\left(x\right)=\left(I_{a+}^{\alpha }f\right)\left(x\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}\frac{{(x-a)}^{\alpha +\beta -1}}{{(b-a)}^{1-\gamma }}{}_2{F}_1\left(\beta ,1-\gamma ;\alpha +\beta ;\frac{x-a}{b-a}\right),$$

(99)

where $a<x<b$, $\alpha >0$, $\beta >0$, $\gamma \in \mathbb{R}$. Here, ${}_2{F}_1(a,b;c;z)$ is the Gauss hypergeometric function, which is defined in the unit disk ($\left|z\right|<1$) as

$${}_2{F}_1(a,b;c;z)=\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k{\left(b\right)}_k}{{\left(c\right)}_k}\frac{z^k}{k!},$$

(100)

where $\left|z\right|<1$, and ${\left(a\right)}_k$ is the Pochhammer symbol. One can see that

$$0<\frac{x-a}{b-a}<1,$$

(101)

for $a<x<b$.

For the case $x=b$, one can obtain

$${}_2{F}_1\left(\beta ,1-\gamma ;\alpha +\beta ;1\right)=\frac{\mathrm{\Gamma }(\alpha +\gamma -1)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(\alpha +\beta +\gamma -1)},$$

(102)

where Equation 1.6.9 in [4], p. 28, is used in the form:

$${}_2{F}_1(a,b;c;1)=\frac{\mathrm{\Gamma }(c-a-b)}{\mathrm{\Gamma }(c-a)\mathrm{\Gamma }(c-b)}.$$

(103)

Note that

$${}_2{F}_1(a,b;c;0)=1.$$

(104)

Using (102), one can obtain

$$F\left(b\right)=\left(I_{a+}^{\alpha }f\right)\left(b\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}\frac{{(b-a)}^{\alpha +\beta -1}}{{(b-a)}^{1-\gamma }}{}_2{F}_1\left(\beta ,1-\gamma ;\alpha +\beta ;1\right)=$$

$$\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{(b-a)}^{\alpha +\beta +\gamma -2}\frac{\mathrm{\Gamma }(\alpha +\gamma -1)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(\alpha +\beta +\gamma -1)}.$$

(105)

As a result, one can define the fractional distribution of the two-parameter power-law type, which is described by the following functions.

The fractional PDF is defined as

$$f_{\alpha }\left(x\right)=\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }\left(\beta \right)}\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(\alpha +\beta +\gamma -1)}{\mathrm{\Gamma }(\alpha +\gamma -1)}\frac{{(x-a)}^{\beta -1}{(b-x)}^{\gamma -1}}{{(b-a)}^{\alpha +\beta +\gamma -2}}.$$

(106)

The fractional CDF is defined as

$$F_{\alpha }\left(x\right)={\left(\frac{x-a}{b-a}\right)}^{\alpha +\beta -1}\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(\alpha +{\beta }_{\gamma }-1)}{\mathrm{\Gamma }(\alpha +\gamma -1)}{}_2{F}_1\left(\beta ,1-\gamma ;\alpha +\beta ;\frac{x-a}{b-a}\right).$$

(107)

One can see that the equalities:

$$F_{\alpha }\left(a\right)=0,$$

(108)

$$F_{\alpha }\left(b\right)=1$$

(109)

are satisfied, if $\alpha +\beta -1>0$, where $a<x<b$, $\alpha >0$, $\beta >0$, and $\gamma \in \mathbb{R}$.

4.3. Fractional Distributions of Confluent Hypergeometric Kummer and Exponential Types

For the function:

$$f\left(x\right)={(x-a)}^{\beta -1}{e}^{\lambda x},$$

(110)

where $x>a$, one can obtain (see Equation 9 in Table 9.1 of [1], p. 173) the equation:

$$F\left(x\right)=\left(I_{a+}^{\alpha }f\right)\left(x\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{e}^{\lambda x}{(x-a)}^{\alpha +\beta -1}{}_1{F}_1\left(\beta ;\alpha +\beta ;\lambda (x-a)\right),$$

(111)

where $\alpha >0$, $\beta >0$, $\lambda \in \mathbb{R}$. Here, ${}_1{F}_1(a;c;z)$ is the confluent hypergeometric Kummer function that is defined by the series:

$${}_1{F}_1(a;c;z)=\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k}{{\left(c\right)}_k}\frac{z^k}{k!},$$

(112)

where ${\left(a\right)}_k$ is the Pochhammer symbol. Note that using ${\left(a\right)}_0=1$, Equation (112) gives

$${}_1{F}_1(a;c;0)=1.$$

(113)

For $x=b$, one can obtain

$$F\left(b\right)=\left(I_{a+}^{\alpha }f\right)\left(b\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{e}^{\lambda b}{(b-a)}^{\alpha +\beta -1}{}_1{F}_1\left(\beta ;\alpha +\beta ;\lambda (b-a)\right).$$

(114)

As a result, one can define the fractional distribution of the confluent hypergeometric Kummer type, which is described by the following functions.

The fractional PDF is defined as

$$f_{\alpha }\left(x\right)=\frac{f\left(x\right)}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}=\frac{1}{F\left(b\right)}{(x-a)}^{\beta -1}{e}^{\lambda x}=$$

$$\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }\left(\beta \right)}\frac{1}{{}_1{F}_1\left(\beta ;\alpha +\beta ;\lambda (b-a)\right)}{e}^{\lambda (x-b)}\frac{{(x-a)}^{\beta -1}}{{(b-a)}^{\alpha +\beta -1}}.$$

(115)

The fractional CDF is defined as

$$F_{\alpha }\left(x\right)=e^{\lambda (x-b)}{\left(\frac{x-a}{b-a}\right)}^{\alpha +\beta -1}\frac{{}_1{F}_1\left(\beta ;\alpha +\beta ;\lambda (x-a)\right)}{{}_1{F}_1\left(\beta ;\alpha +\beta ;\lambda (b-a)\right)},$$

(116)

where $\alpha +\beta -1>0$.

One can see that the equalities:

$$F_{\alpha }\left(a\right)=0,$$

(117)

$$F_{\alpha }\left(b\right)=1$$

(118)

are satisfied, if $\alpha +\beta -1>0$, where $\alpha >0$, $\beta >0$, and $\lambda \mathbb{R}$.

For the case $\beta =1$, the confluent hypergeometric Kummer type gives the fractional distribution of the exponential type, which is described by the following functions.

For function $f\left(x\right)=e^{\lambda x}$, the function can be presented (see Equation 9 in Table 9.1 of [1], p. 173) in the form:

$$F\left(x\right)=\left(I_{a+}^{\alpha }f\right)\left(x\right)=\frac{1}{\mathrm{\Gamma }(\alpha +1)}{e}^{\lambda x}{(x-a)}^{\alpha }{}_1{F}_1\left(1;\alpha +1;\lambda (x-a)\right)=$$

$$e^{\lambda a}{(x-a)}^{\alpha }{E}_{1,\alpha +1}\left(\lambda (x-a)\right)=\frac{e^{\lambda x}}{{\lambda }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\gamma (\alpha ,\lambda (x-a)),$$

(119)

where the equation ${}_1{F}_1(1;\gamma ;z)=\mathrm{\Gamma }\left(\gamma \right)E_{1,\gamma \left(z\right)}$ is used.

Then, The fractional PDF is defined as

$$f_{\alpha }\left(x\right)=\frac{f\left(x\right)}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}=\frac{\mathrm{\Gamma }(\alpha +1)}{{}_1{F}_1\left(1;\alpha +1;\lambda (b-a)\right)}\frac{1}{{(b-a)}^{\alpha }}{e}^{\lambda (x-b)}.$$

(120)

The fractional CDF is defined as

$$F_{\alpha }\left(x\right)={\left(\frac{x-a}{b-a}\right)}^{\alpha }\frac{E_{1,\alpha +1}\left(\lambda (x-a)\right)}{E_{1,\alpha +1}\left(\lambda (b-a)\right)}=e^{\lambda (x-b)}\frac{\gamma (\alpha ,\lambda (x-a\left)\right)}{\gamma (\alpha ,\lambda (b-a\left)\right)},$$

(121)

where $\alpha >0$.

4.4. Fractional Distribution of Gauss Hypergeometric Type

For the function:

$$f\left(x\right)={(x-a)}^{\beta -1}{}_2{F}_1\left(\mu ,\nu ;\beta ;\lambda (x-a)\right),$$

(122)

where $x>a$ and $\lambda \ne 0$, one can obtain (see Equation 23 in Table 9.1 of [1], p. 173) the equation:

$$F\left(x\right)=\left(I_{a+}^{\alpha }f\right)\left(x\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{(x-a)}^{\alpha +\beta -1}{}_2{F}_1\left(\mu ,\nu ;\alpha +\beta ;\lambda (x-a)\right),$$

(123)

where $\alpha >0$, $\beta >0$, and $\lambda \in \mathbb{R}$.

For $x=b$, one can obtain:

$$F\left(b\right)=\left(I_{a+}^{\alpha }f\right)\left(b\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{(b-a)}^{\alpha +\beta -1}{}_2{F}_1\left(\mu ,\nu ;\alpha +\beta ;\lambda (b-a)\right),$$

(124)

where $F\left(x\right)$ and $F\left(b\right)$ is defined in the unit disk

$$\lambda (b-a)<1.$$

(125)

As a result, one can define the fractional distribution of the Gauss hypergeometric type, which is described by the following functions.

The fractional PDF is defined as

$$f_{\alpha }\left(x\right)=\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }\left(\beta \right)}\frac{{(x-a)}^{\beta -1}}{{(b-a)}^{\alpha +\beta -1}}\frac{{}_2{F}_1\left(\mu ,\nu ;\beta ;\lambda (x-a)\right)}{{}_2{F}_1\left(\mu ,\nu ;\alpha +\beta ;\lambda (b-a)\right)}.$$

(126)

The fractional CDF is defined as

$$F_{\alpha }\left(x\right)=\frac{F\left(x\right)}{F\left(b\right)}={\left(\frac{x-a}{b-a}\right)}^{\alpha +\beta -1}\frac{{}_2{F}_1\left(\mu ,\nu ;\alpha +\beta ;\lambda (x-a)\right)}{{}_2{F}_1\left(\mu ,\nu ;\alpha +\beta ;\lambda (b-a)\right)}.$$

(127)

One can see that the equalities:

$$F_{\alpha }\left(a\right)=0,$$

(128)

$$F_{\alpha }\left(b\right)=1$$

(129)

are satisfied, if $\alpha +\beta -1>0$ and $\lambda (b-a)<1$, where $\alpha >0$, $\beta >0$, $\lambda \in \mathbb{R}$.

4.5. Fractional Distribution of Bessel Type

For the function:

$$f\left(x\right)={(x-a)}^{\nu /2}{J}_{\nu }\left(\lambda \sqrt{x-a}\right),$$

(130)

where $x>a$ and $\nu >-1$, one can obtain (see Equation 19 in Table 9.1 of [1], p. 173) the equation:

$$F\left(x\right)=\left(I_{a+}^{\alpha }f\right)\left(x\right)={\left(\frac{2}{\lambda }\right)}^{\alpha }{(x-a)}^{(\nu +\alpha )/2}{J}_{\nu +\alpha }\left(\lambda \sqrt{x-a}\right).$$

(131)

Here, $J_{\nu }\left(z\right)$ is the Bessel function of the first kind, which is defined [4], pp. 32–33, by the equation:

$$J_{\nu }\left(z\right)=\sum _{k=0}^{\infty }\frac{{(-1)}^k}{k!\mathrm{\Gamma }(\nu +k+1)}{\left(\frac{z}{2}\right)}^{2k+\nu },$$

(132)

where $z>0$ and $\nu \in \mathbb{R}$.

Note that

$$J_{-1/2}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{1/2}cos\left(z\right),$$

(133)

$$J_{1/2}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{1/2}sin\left(z\right).$$

(134)

For $x=b$, Equation (131) has the form:

$$F\left(b\right)=\left(I_{a+}^{\alpha }f\right)\left(b\right)={\left(\frac{2}{\lambda }\right)}^{\alpha }{(b-a)}^{(\nu +\alpha )/2}{J}_{\nu +\alpha }\left(\lambda \sqrt{b-a}\right).$$

(135)

The Bessel functions have the following asymptotic form:

$$J_{\nu }\left(z\right)\sim \frac{1}{\mathrm{\Gamma }(\nu +1)}{\left(\frac{z}{2}\right)}^{\nu }(x\to 0+)$$

(136)

for $\nu \ne -1,-2,\dots$. Therefore, for $x\to a+$, Equation (131) gives

$$\underset{x\to a+}{lim}F\left(x\right)=\underset{x\to a+}{lim}\left(I_{a+}^{\alpha }f\right)\left(x\right)=0,$$

(137)

if $\nu +\alpha >0$, where $\nu >-1$ and $\alpha >0$.

As a result, the fractional distribution of the Bessel type is described by the functions.

The fractional PDF is defined as

$$f_{\alpha }\left(x\right)=\frac{1}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}{f}_{\alpha }\left(x\right)=\frac{{(x-a)}^{\nu /2}}{{(b-a)}^{(\nu +\alpha )/2}}\frac{J_{\nu }\left(\lambda \sqrt{x-a}\right)}{J_{\nu +\alpha }\left(\lambda \sqrt{b-a}\right)}.$$

(138)

The fractional CDF is defined as

$$F_{\alpha }\left(x\right)={\left(\frac{x-a}{b-a}\right)}^{(\nu +\alpha )/2}\frac{J_{\nu +\alpha }\left(\lambda \sqrt{x-a}\right)}{J_{\nu +\alpha }\left(\lambda \sqrt{b-a}\right)}.$$

(139)

One can see that the equalities:

$$F_{\alpha }\left(a\right)=0,$$

(140)

$$F_{\alpha }\left(b\right)=1$$

(141)

are satisfied, if $\alpha +\nu >0$, $\alpha >0$, and $\nu >-1$.

4.6. Fractional Distribution of Two-Parameter Mittag–Leffler Type

For the function:

$$f\left(x\right)={(x-a)}^{\beta -1}{E}_{\mu ,\beta }\left({(x-a)}^{\mu }\right),$$

(142)

where $x>a$, $\beta >0$, and $\mu >0$, one can obtain (see Equation 23 in Table 9.1 of [1], p. 173) the equation:

$$F\left(x\right)=\left(I_{a+}^{\alpha }f\right)\left(x\right)={(x-a)}^{\alpha +\beta -1}{E}_{\mu ,\alpha +\beta }\left({(x-a)}^{\mu }\right).$$

(143)

Here, $E_{\mu ,\beta }\left(z\right)$ is the two-parameter Mittag–Leffler function [4], pp. 42–45, of the form:

$$E_{\mu ,\beta }\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\mathrm{\Gamma }(\mu k+\beta )},$$

(144)

where $z\in \mathbb{R}$, $\mu >0$, and $\beta \in \mathbb{R}$.

Note that

$$E_{1,1}\left(z\right)=e^z,E_{1/2}\left(z\right)=exp\left(z^2\right)erfc(-z),$$

(145)

$$E_{1,2}\left(z\right)=\frac{e^z-1}{z},E_{2,1}\left(z\right)=cosh\sqrt{z}$$

(146)

$$E_{2,2}\left(z\right)=\frac{sinh\sqrt{z}}{\sqrt{z}}.$$

(147)

Using $E_{\mu ,\beta }\left(0\right)=1/\mathrm{\Gamma }\left(\beta \right)$, for $x=a$, one can obtain

$$F\left(a\right)=\left(I_{a+}^{\alpha }f\right)\left(b\right)=0$$

(148)

For $x=b$, Equation (143) gives

$$F\left(b\right)=\left(I_{a+}^{\alpha }f\right)\left(b\right)={(b-a)}^{\alpha +\beta -1}{E}_{\mu ,\alpha +\beta }\left({(b-a)}^{\mu }\right).$$

(149)

As a result, the fractional distribution of the two-parameter Mittag–Leffler type is defined by the functions.

The fractional PDF is defined as

$$f_{\alpha }\left(x\right)=\frac{1}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}{f}_{\alpha }\left(x\right)=\frac{{(x-a)}^{\beta -1}}{{(b-a)}^{\alpha +\beta -1}}\frac{E_{\mu ,\beta }\left({(x-a)}^{\mu }\right)}{E_{\mu ,\alpha +\beta }\left({(b-a)}^{\mu }\right)}.$$

(150)

The fractional CDF is defined as

$$F_{\alpha }\left(x\right)={\left(\frac{x-a}{b-a}\right)}^{\alpha +\beta -1}\frac{E_{\mu ,\alpha +\beta }\left({(x-a)}^{\mu }\right)}{E_{\mu ,\alpha +\beta }\left({(b-a)}^{\mu }\right)}.$$

(151)

One can see that the equalities:

$$F_{\alpha }\left(a\right)=0,$$

(152)

$$F_{\alpha }\left(b\right)=1$$

(153)

are satisfied, if $\alpha +\beta -1>0$, where $\mu >0$, $\alpha >0$, and $\beta \in \mathbb{R}$.

4.7. Fractional Distribution of Prabhakar Type

Let us consider the function:

$$f\left(x\right)={(x-a)}^{\beta -1}{E}_{\mu ,\beta }^{\gamma }[-\eta {(x-a)}^{\mu }],$$

(154)

where $\beta >0$, $\mu >0$, $\eta \in \mathbb{R}$, $x\in [a,b]$, and $E_{\mu ,\beta }^{\gamma }\left[z\right]$ is the Prabhakar function [62,63,64,65,66,67]. The Prabhakar function (see Section 5.1 in [68], pp. 115–128) is defined as

$$E_{\mu ,\beta }^{\gamma }\left[x\right]=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_n}{n!\mathrm{\Gamma }(\beta +\mu n)}{x}^n,$$

(155)

where $\mu >0$, $\beta >0$, $\gamma >0$, and ${\left(\gamma \right)}_n=\mathrm{\Gamma }(\gamma +n)/\mathrm{\Gamma }\left(\gamma \right)$ is the Pochhammer symbol for $n\in \mathbb{N}$.

Using the equation in the form:

$${\int }_0^x\frac{{(x-u)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{u}^{\beta -1}{E}_{\mu ,\beta }^{\gamma }[-\eta u^{\mu }]du=x^{\beta +\alpha -1}{E}_{\mu ,\beta +\alpha }^{\gamma }[-\eta x^{\mu }],$$

(156)

which was proven in Theorem 5.5 in [68], p. 125, where $\alpha >0$, $\beta >0$, $\mu >0$, $\eta \in \mathbb{R}$, and the variable $z=u-a$, one can obtain

$${\int }_a^x\frac{{(x-u)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{(u-a)}^{\beta -1}{E}_{\mu ,\beta }^{\gamma }[-\eta {(u-a)}^{\mu }]du=$$

$${\int }_0^{x-a}\frac{{((x-a)-z)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{z}^{\beta -1}{E}_{\mu ,\beta }^{\gamma }[-\eta z^{\mu }]dz={(x-a)}^{\beta +\alpha -1}{E}_{\mu ,\beta +\alpha }^{\gamma }[-\eta {(x-a)}^{\mu }].$$

(157)

Then, the function:

$$F\left(x\right)=\left(I_{a+}^{\alpha }f\right)\left(x\right)={\int }_a^x\frac{{(x-u)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(u\right)du,$$

(158)

where $f\left(x\right)$ is given by Equation (154) takes the form:

$$F\left(x\right)={(x-a)}^{\beta +\alpha -1}{E}_{\mu ,\beta +\alpha }^{\gamma }[-\eta {(x-a)}^{\mu }].$$

(159)

Equation (155) gives

$$F\left(x\right)\sim \frac{1}{\mathrm{\Gamma }\left(\beta \right)}{(x-a)}^{\beta +\alpha -1}(x\to a+).$$

(160)

Therefore, the condition $F(a+)=0$ holds, if

$$\alpha +\beta -1>0.$$

(161)

Using Equation 5.1.31 of Theorem 5.4 in [68], p. 121, (see also [66]), one can consider the asymptotic expansion for $x\to \infty$. This expansion gives the restriction $\beta +\alpha -1-\mu \gamma >0$, if the interval is infinite $[0,\infty$ [28]. For the finite interval $[a,b]\subset \mathbb{R}$, this restriction does not apply.

As a result, the fractional distribution of the Prabhakar type is described by the functions as follows.

The fractional PDF is defined by the equation:

$$f_{\alpha }\left(x\right)=\frac{1}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}f\left(x\right)=\frac{{(x-a)}^{\beta -1}}{{(b-a)}^{\beta +\alpha -1}}\frac{E_{\mu ,\beta }^{\gamma }[-\eta {(x-a)}^{\mu }]}{E_{\mu ,\beta +\alpha }^{\gamma }[-\eta {(b-a)}^{\mu }]}.$$

(162)

The fractional CDF is defined as

$$F_{\alpha }\left(x\right)={\left(\frac{x-a}{b-a}\right)}^{\beta +\alpha -1}\frac{E_{\mu ,\beta +\alpha }^{\gamma }[-\eta {(x-a)}^{\mu }]}{E_{\mu ,\beta +\alpha }^{\gamma }[-\eta {(b-a)}^{\mu }]}.$$

(163)

One can see that the equalities:

$$F_{\alpha }\left(a\right)=0,$$

(164)

$$F_{\alpha }\left(b\right)=1$$

(165)

are satisfied, if $\alpha +\beta -1>0$, where $\alpha >0$$\beta >0$, $\mu >0$, and $\eta \in \mathbb{R}$.

5. Fractional Mean (Average) Values

In this section, fractional generalizations of the mean (average) values are suggested for fractional probability theory.

Let $A\left(X\right)$ be a function of a random variable X, which is distributed with a probability density $f_1\left(x\right)$ on the interval $[a,b]$. Then, the standard average value of $A\left(X\right)$ is described as

$$\langle A\left(X\right)\rangle =\mathsf{E}\left[A\left(X\right)\right]:={\int }_a^{b}A\left(x\right)f_1\left(x\right)dx.$$

(166)

Using the standard CDF:

$$F_1\left(x\right)={\int }_a^x{f}_1\left(u\right)du,$$

(167)

Equation (166) can be given as

$$\langle A\left(X\right)\rangle =\mathsf{E}\left[A\left(X\right)\right]:={\int }_a^{b}A\left(x\right)d{F}_1\left(x\right),$$

(168)

where $d{F}_1\left(x\right)=F_1^{\left(1\right)}\left(x\right)dx=f_1\left(x\right)dx$.

5.1. Definition of Fractional Average (Mean) Values

To define a fractional generalization of the standard Equations (166) and (168), the following corresponding principles should be taken into account.

The first corresponding principle describes the fractional normalization condition as the average value of the unit function of a random variable:

$${\langle A\left(X\right)\rangle }_{\alpha }={\mathsf{E}}_{\alpha }\left[A\left(X\right)\right]=1,$$

(169)

if $A\left(X\right)=1$ for $x\in [a,b]$ and $\alpha >0$.

The second corresponding principle means that the fractional average value of order $\alpha >0$ should be equal to the standard average value for $\alpha =1$:

$${\langle A\left(X\right)\rangle }_{\alpha =1}={\mathsf{E}}_{\alpha =1}\left[A\left(X\right)\right]={\int }_a^{b}A\left(x\right)d{F}_1\left(x\right)={\int }_a^{b}A\left(x\right)f_1\left(x\right)dx.$$

(170)

Let us define fractional average values of function $A\left(X\right)$, which is distributed with a fractional probability density $f_{\alpha }\left(x\right)$ on the interval $[a,b]$.

Definition 13. 

(Fractional average values of function $A\left(X\right)$)

Let $f_{\alpha }\left(x\right)$ be a fractional probability density function, $A\left(X\right)$ be a function of a random variable X, and $F_{\alpha }\left(x\right)$ be the fractional cumulative distribution function such that

$$F_{\alpha }\left(x\right)=\left(I_{a+}^{\alpha }{f}_{\alpha }\right)\left(x\right).$$

(171)

Let $A\left(x\right)f_{\alpha }\left(x\right)\in L_p(a,b)$. Then, the value:

$${\langle A\left(X\right)\rangle }_{\alpha }={\mathsf{E}}_{\alpha }\left[A\left(X\right)\right]:=\left(I_{a+}^{\alpha }A{f}_{\alpha }\right)\left(x\right)={\int }_a^b\frac{{(b-x)}^{\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}A\left(x\right)f_{\alpha }\left(x\right)dx$$

(172)

is called the fractional average value of the first type.

Let $A\left(x\right)F_{\alpha }^{\left(1\right)}\left(x\right)\in L_p(a,b)$. Then, the value:

$${\langle A\left(X\right)\rangle }_{\alpha }^{*}={\mathsf{E}}_{\alpha }^{*}\left[A\left(X\right)\right]:=\left(I_{a+}^{\alpha }A{F}_{\alpha }^{\left(1\right)}\right)\left(x\right)={\int }_a^b\frac{{(b-x)}^{\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}A\left(x\right)d{F}_{\alpha }\left(x\right),$$

(173)

where $F_{\alpha }^{\left(1\right)}\left(x\right)=d{F}_{\alpha }\left(x\right)/dx$, is called the fractional average value of the second type.

Note that, in the general case, the fractional average values of the first and second types do not coincide.

$${\langle A\left(X\right)\rangle }_{\alpha }\ne {\langle A\left(X\right)\rangle }_{\alpha }^{*}.$$

(174)

These average values are only equivalent if $\alpha$ is equal to one, i.e., ${\langle A\left(X\right)\rangle }_1\ne {\langle A\left(X\right)\rangle }_1^{*}$.

5.2. Fractional Average Values for Fractional Distributions of Power-Law and Uniform Types

Let us consider the fractional average (mean) value of the function $A\left(x\right)={(x-a)}^m$ with $m>0$ for the fractional distributions of the power-law and uniform types.

For the function:

$$A\left(x\right)f\left(x\right)={(x-a)}^{\beta +m-1},$$

(175)

where $x>a$ and $\beta >0$, one can obtain (see Equation 1 in Table 9.1 of [1], p. 173) the equation:

$$\left(I_{a+}^{\alpha }Af\right)\left(x\right)=\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}{(x-a)}^{\alpha +\beta +m-1}.$$

(176)

For $x=b$, Equation (85) has the form:

$$\left(I_{a+}^{\alpha }f\right)\left(b\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{(b-a)}^{\alpha +\beta -1},$$

(177)

$$\left(I_{a+}^{\alpha }Af\right)\left(b\right)=\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}{(b-a)}^{\alpha +\beta +m-1}.$$

(178)

To obtain the fractional average (mean) value of the first type, one can use the equation:

$${\langle A\left(X\right)\rangle }_{\alpha }=\frac{1}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}\left(I_{a+}^{\alpha }Af\right)\left(b\right).$$

(179)

As a result, for $A\left(x\right)={(x-a)}^m$, one can obtain the fractional average value of the first type:

$${\langle A\left(X\right)\rangle }_{\alpha }={\mathsf{E}}_{\alpha }\left[{(x-a)}^m\right]:=\left(I_{a+}^{\alpha }A{f}_{\alpha }\right)\left(x\right)\underset{x\to b-}{lim}\left(I_{a+}^{\alpha }A{f}_{\alpha }\right)\left(x\right)=$$

$$\frac{1}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}\underset{x\to b-}{lim}\left(I_{a+}^{\alpha }Af\right)\left(x\right)=\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }\left(\beta \right)}{(b-a)}^m$$

(180)

For the case $\beta =1$, the fractional average value (180) for the fractional distribution of the power-law type gives the fractional average value for the fractional distribution of the uniform type in the form:

$${\langle {(x-a)}^m\rangle }_{\alpha }={\mathsf{E}}_{\alpha }\left[{(x-a)}^m\right]:=\frac{\mathrm{\Gamma }(m+1)\mathrm{\Gamma }(\alpha +1)}{\mathrm{\Gamma }(\alpha +m+1)}{(b-a)}^m,$$

(181)

where $\alpha >0$.

Using the fractional CDF of the fractional distribution of the power-law type in the form:

$$F_{\alpha }\left(x\right)={\left(\frac{x-a}{b-a}\right)}^{\alpha +\beta -1},$$

(182)

where $\alpha >0$, $\beta >0$, and $\alpha +\beta -1>0$, one can obtain

$$F_{\alpha }^{\left(1\right)}\left(x\right)=\frac{\alpha +\beta -1}{{(b-a)}^{\alpha +\beta -1}}{(x-a)}^{\alpha +\beta -2},$$

(183)

and

$$A\left(x\right)F_{\alpha }^{\left(1\right)}\left(x\right)=\frac{\alpha +\beta -1}{{(b-a)}^{\alpha +\beta -1}}{(x-a)}^{\alpha +\beta +m-2}.$$

(184)

As a result, the fractional average value of the second type for the function $A\left(x\right)={(x-a)}^m$ and the fractional distribution of the power-law type takes the form:

$${\langle A\left(X\right)\rangle }_{\alpha }^{*}=\left(I_{a+1}^{\alpha }A{F}_{\alpha }^{\left(1\right)}\right)\left(x\right)=\frac{\alpha +\beta -1}{{(b-a)}^{\alpha +\beta -1}}\frac{\mathrm{\Gamma }(\alpha +\beta +m-1)}{\mathrm{\Gamma }(2\alpha +\beta +m-1)}{(x-a)}^{2\alpha +\beta +m-2}.$$

(185)

5.3. Fractional Average Values for Fractional Distribution of Two-Parameter Power-Law Type

For the product:

$$A\left(x\right)f\left(x\right)={(x-a)}^{\beta +m-1}{(b-x)}^{\gamma -1},$$

(186)

one can obtain (see Equation 3 in Table 9.1 of [1], p. 173) the equation:

$$\left(I_{a+}^{\alpha }Af\right)\left(x\right)=\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}\frac{{(x-a)}^{\alpha +\beta +m-1}}{{(b-a)}^{1-\gamma }}{}_2{F}_1\left(\beta +m,1-\gamma ;\alpha +\beta +m;\frac{x-a}{b-a}\right),$$

(187)

where $a<x<b$, $\alpha +\beta -1>0$, $\alpha >0$, $\beta >0$, and $\gamma \in \mathbb{R}$.

Using (102), one can obtain

$$\left(I_{a+}^{\alpha }f\right)\left(b\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}\frac{{(b-a)}^{\alpha +\beta -1}}{{(b-a)}^{1-\gamma }}{}_2{F}_1\left(\beta ,1-\gamma ;\alpha +\beta ;1\right)=$$

$$\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}\frac{\mathrm{\Gamma }(\alpha +\gamma -1)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(\alpha +\beta +\gamma -1)}{(b-a)}^{\alpha +\beta +\gamma -2},$$

(188)

and

$$\left(I_{a+}^{\alpha }Af\right)\left(b\right)=\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}\frac{{(b-a)}^{\alpha +\beta +m-1}}{{(b-a)}^{1-\gamma }}{}_2{F}_1\left(\beta +m,1-\gamma ;\alpha +\beta +m;1\right)=$$

$$\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}\frac{\mathrm{\Gamma }(\alpha +\gamma -1)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(\alpha +\beta +m+\gamma -1)}{(b-a)}^{\alpha +\beta +m+\gamma -2}.$$

(189)

As a result, $A\left(x\right)={(x-a)}^m$, and one can obtain

$${\langle A\left(X\right)\rangle }_{\alpha }={\mathsf{E}}_{\alpha }\left[{(x-a)}^m\right]:=\left(I_{a+}^{\alpha }A{f}_{\alpha }\right)\left(x\right)=\frac{\left(I_{a+}^{\alpha }Af\right)\left(b\right)}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}=$$

$$\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }\left(\beta \right)}\frac{\mathrm{\Gamma }(\alpha +\beta +\gamma -1)}{\mathrm{\Gamma }(\alpha +\beta +m+\gamma -1)}{(b-a)}^m.$$

(190)

Here, $\alpha +\beta -1>0$, where $a<x<b$, $\alpha >0$, $\beta >0$, and $\gamma \in \mathbb{R}$.

5.4. Fractional Average Values for Fractional Distribution of Confluent Hypergeometric Kummer and Exponential Types

For the product of function:

$$A\left(x\right)f\left(x\right)={(x-a)}^{\beta +m-1}{e}^{\lambda x},$$

(191)

where $x>a$, one can obtain (see Equation 9 in Table 9.1 of [1], p. 173) the equation:

$$\left(I_{a+}^{\alpha }Af\right)\left(x\right)=\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}{e}^{\lambda x}{(x-a)}^{\alpha +\beta +m-1}{}_1{F}_1\left(\beta +m;\alpha +\beta +m;\lambda (x-a)\right),$$

(192)

where $\alpha >0$, $\beta >0$, $\lambda \in \mathbb{R}$.

For $x=b$, one can obtain

$$\left(I_{a+}^{\alpha }f\right)\left(b\right)=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{e}^{\lambda b}{(b-a)}^{\alpha +\beta -1}{}_1{F}_1\left(\beta ;\alpha +\beta ;\lambda (b-a)\right),$$

(193)

and

$$\left(I_{a+}^{\alpha }Af\right)\left(b\right)=\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}{e}^{\lambda b}{(b-a)}^{\alpha +\beta +m-1}{}_1{F}_1\left(\beta +m;\alpha +\beta +m;\lambda (b-a)\right).$$

(194)

As a result, for $A\left(x\right)={(x-a)}^m$, one can obtain

$${\langle A\left(X\right)\rangle }_{\alpha }={\mathsf{E}}_{\alpha }\left[{(x-a)}^m\right]:=\left(I_{a+}^{\alpha }A{f}_{\alpha }\right)\left(x\right)=\frac{\left(I_{a+}^{\alpha }Af\right)\left(b\right)}{\left(I_{a+}^{\alpha }f\right)\left(b\right)}=$$

$$\frac{\mathrm{\Gamma }(\beta +m)}{\mathrm{\Gamma }(\alpha +\beta +m)}\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }\left(\beta \right)}\frac{{}_1{F}_1\left(\beta +m;\alpha +\beta +m;\lambda (b-a)\right)}{{}_1{F}_1\left(\beta ;\alpha +\beta ;\lambda (b-a)\right)}{(b-a)}^m.$$

(195)

For the case $\beta =1$, the fractional average value (195) type gives the fractional distribution of the exponential type.

6. Conclusions

A generalization of probability theory was formulated by applying fractional integrals and fractional derivatives of arbitrary (non-integer and integer) orders.

Briefly, we list the main results obtained in the article.

• The definition of the fractional probability density function (fractional PDF) was proposed. The basic properties of the fractional PDF were proven.
• The definition of the fractional cumulative distribution function (fractional CDF) was suggested, and the basic properties of these functions were also proven.
• Using the properties of the fractional CDF, two theorems, which prove the existence of the probability space of the fractional probability of arbitrary order $\alpha >0$, were suggested.
• Examples of the distributions of the fractional probability on finite intervals $[a,b]$ were proposed. The following distributions of the fractional probability (DFP, fractional distributions) were considered: (a) The DFP of the power-law type; (b) The DFP of the uniform type; (c) The DFP of the two-parameter power-law type; (d) The DFP of the confluent hypergeometric Kummer type; (e) The DFP of the exponential type; (f) The DFP of the Gauss hypergeometric type; (g) The DFP of the Bessel type; (h) The DFP of the two-parameter Mittag–Leffler type; (i) The DFP of the Prabhakar type.
• Fractional average (mean) values were defined, and examples of the calculations were suggested.

Some promising generalizations of the proposed approach to the fractional probability theory should be noted:

(A) The proposed results can be generalized to a high dimension on $R^d$ and bounded domains in $R^d$ with integer $d>0$ by analogy with the work [28]. To realize this generalization, partial and mixed fractional integrals and fractional derivatives ([4], pp. 123–127) and the results of general fractional vector calculus [69] can be used. It should be noted that the violation of the standard chain rule leads to the fact that fractional differential and integral operators, which are defined in different coordinate systems (Cartesian, cylindrical, and spherical), are not related to each other by coordinate transformations. Therefore, the definitions of the fractional d-dimensional operators should be formulated separately in spherical, cylindrical, and other coordinates. In this case, the operators themselves can be defined in orthogonal curvilinear coordinates (OCC) using the Lame coefficients. Equations written in terms of Lame coefficients will define these operators for all orthogonal coordinate systems, but these expressions are not related to each other by the coordinate transformations. Methods for describing piecewise simple regions in general fractional calculus for a wider class of domains, surfaces, and curves were described in the paper [69]. Similarly, bounded domains in $R^d$ can be considered in the fractional calculus and the fractional probability theory.

(B) The fractional probability function can be generalized from the Riemann–Liouville fractional integrals, and derivatives can be generalized to using the Riesz fractional integral and derivatives ([4,70,71,72] and Sections 25 and 26 of [1]). It should be noted that the Riesz type of the d-dimensional fractional operators is mainly adequate for the case of the spherical symmetry in d-dimensional space. This is due to the obvious fact that the symmetry of the operators used and the coordinate system must be related to the symmetry of the problem. In addition, function spaces (Sections 25 and 26 of [1]), to which the fractional (nonlocal) probability density functions and fractional (nonlocal) cumulative distribution functions must belong, should be defined mathematically correctly and be mutually consistent.

(C) To generalize the proposed results for the wide class of operator kernels, it is important to expand the Luchko form of the general fractional calculus (see [48,49,50,51,52,53,54]) for the distribution on finite intervals $[a,b]$. Unfortunately, this problem has not yet been solved (see Section 2.4 “Open problems” in [57], p. 3255).