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Appl. Math. Inf. Sci. Lett. 3, No. 2, 61-67 (2015) 61 Applied & Information Sciences Letters An International Journal

http://dx.doi.org/10.12785/amisl/030204

The Generalized Appears the Trace of in

Masoud Ganji1,∗ and Fatemeh Gharari2 1 Department of Statistics, University of Mohaghegh Ardabili, Ardabil, Iran 2 Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran

Received: 5 Aug. 2014, Revised: 14 Mar. 2015, Accepted: 17 Mar. 2015 Published online: 1 May 2015

Abstract: In this paper, introducing a generalized random variable, we obtained a new relationship between fractional calculus and statistics. With this definition, the domain of space for some of distributions, such as Gamma, Beta and Weibull, were expanded from (0,∞) to (−1,∞). It is shown that the expectation of such random variable ,Φα (x), coincides with the fractional of probability density function (PDF), at the origin, for α > 0, and also the fractional of PDF, at the origin, for −1 < α < 0. We also, presented PDFs of such distributions as a product of fractional derivation of Dirac function of shape parameter order, α δ ( )(.). Finally, we showed that the Liouville fractional differintegral on the generating function (MGF) of positive random variable, at the origin, gives fractional moments.

Keywords: Fractional calculus, , Fractional moments, , ,

1 Introduction Dirac delta function which is defined as: δ C The study of generalized functions is now widely used in : X → and sciences. δ(x)ϕ(x)dx = ϕ(0) Generalized functions are defined as a linear functional Z on a space X of conveniently chosen test functions. ϕ For every locally integrable function f L 1 R , there for every test function ∈ X. The test function space X is ∈ loc( ) ∞ R exists a distribution F : X → C defined by: usually chosen as a subspace of C ( ), the space of f infinitely differentiable functions [6]. ∞ In the present work, we suppose that X is a positive Ff (ϕ)= h f ,ϕi = f (x)ϕ(x)dx (1) random variable and α is the shape parameter of ∞ Z− distribution. The new random variable is represented as α X −1 where ϕ ∈ X is test function from a suitable space X of Φα Φα + the function (x) and defined by (x) = Γ (α) . The test functions. A distribution that corresponds to functions function Φα (x) can be extended to all complex values of via equation (1) are called regular distributions. Examples α α as a pseudo function and is a distribution whose for regular distributions are the kernels K± ∈ is [0,∞) except for the case α = 0,−1,.... Since L 1 R loc( ) defined as: distributions or generalized functions in mathematical analysis are not really functions in the classical sense, we also call our proposed random variable a generalized α−1 α ±x K = H(±x) (2) random variable. The expectation of this generalized ± Γ (α) random variable coincides with Riemann-Liouville left fractional integral of the PDF, at the origin, for α > 0 and for α > 0 and H(x) is a Heaviside unit step function. Marchaud fractional derivative of the PDF, at the origin, Distributions that are not regular are sometimes called for −1 < α < 0. The generalized random variable singular. An example for a is the appears in some distributions that belong to the

∗ Corresponding author e-mail: [email protected]

c 2015 NSP Natural Sciences Publishing Cor. 62 M. Ganji, F. Gharari: The Generalized Random Variable Appears the Trace...

like Weibull, Gamma, Beta, Binomial where α > −1 and β > −1 such that α + β > −1 and we and poisson distributions. By this generalized random used the star notation for the convolution operation. variable, we more expand the domain of shape parameter The proof is easy for α > 0 and β > 0, as instance, see space. For example, the domain of shape parameter which [2]. Other values of α and β can be proved using analytic has been already expanded for Gamma and Weibull continuation. distributions of α > 0 to α > −1 and for Beta distribution from (0,∞) × (0,∞) to (−1,∞) × (0,∞). In case of (e) If X1,X2,...,Xn be i.i.d. W(x;α,θ) then negative values of the shape parameter space, the Φ Γ Γ α θ relationship between fractional and statistics α+1(x) ∼ (1, ( + 1) ), theory can be obtained, this we can write the PDF of distributions like Gamma, Weibull, Beta as a product and of fractional derivatives of Dirac delta function. The Φα+1(x) characteristics of this generalized random variable are as ∼ B(1,n − 1). Σ nΦα (x ) following: 1 +1 i Also, we demonstrate that the αth moment, α ∈ R, of the (a) By rewriting Binomial and poisson distributions in positive random variable X can be obtained directly from terms of the generalized random variable, they are the Liouville fractional derivation and integrations of the respectively transformed into Gamma and Beta MGF at the origin. distributions with a discrete parameter space, that is

2 Preliminaries [P(x;λ);Φλ (x)] = [Γ (λ;x,1);Φx(λ)], x = 0,1,... and In this section, we introduce notations, definitions and preliminary facts which are used throughout this paper. [Bin(x; p);Φp(x)] = [B(p;x,n − x + 2);Φx(p)] We need some basic definitions and properties of the fractional calculus theory and the generalized functions where x = 1,2,...,n and [.,Φ] is the rewriting form of theory which are used further in this paper. As mentioned generalized random variable. in references [1] and [3], the definitions 1 and 2 of fractional calculus are as following: (b) The expectation of this generalized random Φ variable, α (x), coincides with Riemann-Liouville left Definition 1. For a function f defined on an interval [a,b], fractional integral of the PDF at the origin for α > 0 and α α the Riemann-Liouville (R-L) Ia+ f and Ib− f of Marchaud fractional derivative of the PDF at the origin order α ∈ C , (R(α) > 0) are defined, respectively, by for −1 < α < 0, that is, we have t α 1 α (I f )(t)= (t − ξ ) −1 f (ξ )dξ (7) a+ Γ (α) α α Za Φ (I− f )(0), > 0 E[ α (X)] = α α (3) ((D− f )(0), −1 < < 0 and b α 1 α where (I f )(t)= (ξ −t) −1 f (ξ )dξ . (8) b− Γ (α) ∞ Zt α 1 α−1 (I− f )(x)= t f (x +t)dt (4) α Γ (α) 0 Also, the left and right R-L fractional derivations Da+ f Z α α C R α and Db− f of order ∈ , ( ( ) > 0) are defined, is the Riemann-Liouville left fractional integral, while respectively, by ∞ α 1 α α d α (D f )(x)= t− −1{ f (x +t) − f (x)}dt (5) (D f )(t) = ( )n(In− f )(t), (9) − Γ (−α) a+ dt a+ Z0 is the Marchaud fractional derivative. and α −d n n−α (Db− f )(t) = ( ) (Ia+ f )(t). (10) (c) The integer and fractional derivatives of this dt generalized random variable are the generalized random If in above definition, respectively, a = −∞ and b = ∞, variable, too. then we get Liouville fractional differintegral.

(d) Another property is α Definition 2. The Liouville fractional differintegral D± is defined by Φ Φ Φ α (t − a) ∗ β (t)= α+β (t − a), (6)

c 2015 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. Lett. 3, No. 2, 61-67 (2015) / www.naturalspublishing.com/Journals.asp 63

for 0 < α < 1. [6]

Γ α x −α−1 , ∞ R ′ R+ α frac1 (− ) −∞(x −t) f (t)dt Now if f ∈ C0 ( ) with supp f ⊂ , then D+{ f (x)} = dn α−n ( dxn {D+ { f (Rx)} (11) α α −α 0 δ (α) D0+ f = D0+(I f ) = (K+ · K+) · f = · f (20) R α where the first expression satisfies for ( ) < 0 and for all α ∈ C. Also, the differentiation rule the second expression satisfies for R(α) > 0;n = [R(α)] + 1. Also α β β −α D0+K+ = K+ (21) β α C 1 ∞ holds for all , ∈ . It contains (x −t)−α−1 f (t)dt, α Γ α x D f x (− ) (12) β β −1 −{ ( )} =  n n DK+ = K+ (−1) d R α−n  n {D− { f (x)} dx for all β ∈ C as a special case [6]. with R(α) <0 for the first expression and R(α) > 0 such that n = [R(α)] + 1, for the second expression satisfies. In particular, when α = n ∈ N0, then 3 The generalized Weibull random variable 0 n (n) The Wiebull distribution was originally used for modeling D+{ f (x)} = f (x),D+{ f (x)} = f (x), (13) fatigue data and is at present used extensively in and engineering problems for modeling, the distribution of the lifetime of an object which consists of several parts and 0 n n (n) D−{ f (x)} = f (x),D−{ f (x)} = (−1) f (x). (14) that fails if any component part fails [7]. It is important to note that the fractional calculus has also ∞ R ′ Definition 3. Let f be a f ∈ C0 ( ) been applied in describing several PDFs in mathematical with supp f ⊂ R+. Then its fractional integral is the statistics in terms of fractional integral and derivative of α distribution I0+ f defined as: exponential functions. For instance, the Wiebull

distribution can be writtenα in terms of the fractional α ϕ α ϕ α ϕ −θx hI0+ f , i = hI f , i = hK+ ∗ f , i (15) derivative or integral of e , as

α for R(α) > 0[6]. Also, the fractional derivative of order θ 1−αΓ α Φ α −θt ( + 1) α (x)(D ;xα e )(x), (22) α with lower limit 0 is the distribution Dα { f (z)}defined − as: or α α α α 1+α α −θt ϕ ϕ − ϕ θ Γ α Φα α hD0+ f , i = hD f , i = hK+ f , i (16) ( + 1) (x)(I−;x e )(x), (23) α α where ∈ C and where the subscript x indicates fractional derivative or integral with respect to the variable xα . In this way representation of the PDF might introduce novel xα−1 statistical interpretations in the study of statistical H(x) , R(α) > 0 problems involving such these PDFs (at least in some α Γ (α) K+(x)=  n α+n−1 cases).  d x R α N We will define the Weibull distribution as a  [H(x) ], ( )+ n > 0;n ∈ dxn Γ (α + n) two-parameter family of distribution functions, in which (17) θ α  the parameter > 0 is the scale parameter and −1 < is  the shape parameter. So, this distribution can be defined α is the distribution. For = 0 one can finds as K0 (x) = ( d )H(x) = δ(x) and D0 = I as the identity + dx 0+ 2 α N 2 −Γ (α+1)Φα+1(θ)Φα+1(x) operator. For the = −n; n ∈ , one can finds Γ (α + 1)Φα+1(θ)DΦα+1(x)e . −n δ (n) θ K+ (x)= (x) (18) On the other hand, the scale parameter , can also be considered as a random variable. This means that, where δ (n) is the nth derivative of the δ distribution. The depending on the variable which is being differentiated, kernel distribution in equation (17) is given by: we have the distribution of that random variable. Therefore, the PDF can be written as follows: −α α d x d α 1− 2 K+(x)= [H(x) ]= K+ (x) (19) −Γ (α+1)Φα 1(θ)Φα 1(x) dx Γ (1 − α) dx −Dxe + + . (24)

c 2015 NSP Natural Sciences Publishing Cor. 64 M. Ganji, F. Gharari: The Generalized Random Variable Appears the Trace...

And also, if X be random variable of Wiebull distribution 4 The generalized Gamma random variable as following:

2 −Γ 2(α+1)Φα (θ)Φα (x) In this section, we will define the Gamma distribution as Γ (α + 1)Φα (θ)DΦα (x)e +1 +1 , +1 +1 a two-parameter family of distribution functions, in which (25) the parameter β > 0 is the scale parameter, and −1 < α is by using equation (21) with β = 0 and equation (17), we the shape parameter. This distribution is defined as can rewrite the PDF in terms of the fractional derivation −Φ2(β )Φ2(x) of Dirac delta function, as following definition. Γ (α + 1)Φα+1(β)DΦα+1(x)e ,

Definition 3.1. Suppose that X be a random variable of The scale parameter β, can also be considered as a Weibull distribution as a two-parameter family of random variable. This means that, depending on which distribution functions, in which the parameter θ is the variable is under differentiation, we have the distribution scale parameter and α is the shape parameter. Then, PDF of that random variable; that is, the PDF can be rewritten of this distribution can be defined as as:

α Γ 2 α Φ θ Φ Γ (α + 1)Φα+1(β)DxΦα+1(x)exp − Φ2(β)Φ2(x), (28) Γ 2 α Φ θ − ( +1) α+1( ) α+1(x) ( + 1) α+1( )K+(x)e f (x)= α Γ 2 α Φ θ Φ Γ 2 α Φ θ δ ( ) − (1− ) 1−α ( ) 1−α (x) ( (1 − ) 1−α( ) (x)e or as (26) α −Φ2(β )Φ2(x) where the frist expression satisfies for 0 < and the Γ (α + 1)Dβ Φα+1(β)Φα+1(x)e . (29) second expression satisfies for −1 < α ≤ 0 (For comfortable computationally, we are showing Similarly as a general case of Weibull distribution, if X be −1 < α ≤ 0 with −α.). It can write, for 0 < α < 1, a random variable, by using equation (21) with β = 0 and 2 equation (17), we can write its PDF as following Γ 2 α Φ θ δ (α−1) −Γ (α)Φα (θ)Φα (x) fX (x)= ( ) α ( ) (x)e . definition: ∞ Now we are showing that 0 fX (x)dx = 1, for the case −1 < α ≤ 0, too. We have R Definition 4.1. Suppose that X be a random variable of Gamma distribution as a two-parameter family of distribution functions, in which the parameter β is the ∞ ∞ scale parameter and α is the shape parameter. Then, PDF Γ α θ −α δ (α) fX (x)dx = (1 − ) (x) of this distribution can be defined as Z0 Z0 α −α −Γ (1−α)θ − x × e Γ (1−α) dx, α Φ β 2 Γ α Φ β − 2( )K+(x) ( + 1) α+1( )K+(x)e by using equation (19) fX (x)= 2 (α) −Φ2(β )K (x) ∞ Γ (1 − α)Φ1−α (β)δ (x)e + α α Γ α θ −α δ (α−1) ( = Γ (1 − α)θ − δ ( )(x).e− (1− ) (x)dx, (30) Z0 α under the substitution u = δ ( −1)(x), where the first expression satisfies for 0 < α and the ∞ second expression satisfies for −1 < α ≤ 0. α Γ 1 α θ −α u = Γ (1 − α)θ − e− ( − ) du, It can write, for 0 < α < 1, Z0 −α −1 α = Γ (1 − α)θ .Γ (1 − α)θ , (α−1) −Φ (β )K2 (x) fX (x)= Γ (α)Φα (β)δ (x)e 2 + . which proves our result. Also the failure rate (or hazard function) for distribution is The MGF of the Gamma generalized random variable is as given by following:

β −α β α α Γ 2 α Φ θ α α MX (−t)= (t + ) , − 1 < ≤ 0. (31) ( + 1) α+1( )K+(x), > 0 h(x)= α Γ 2(1 − α)Φ α (θ)δ ( )(x), −1 < α ≤ 0. α α ( 1− Since L{δ ( )(x)} = s , which implied that (27) ∞ α 0 fX (x)dx = 1 for −1 < ≤ 0 and the MGF be as above, such that we have So, we succeed to represent the PDF of Weibull R distribution with the extended shape parameter space, −1 < α, as a product of fractional derivation of Dirac ∞ ∞ −α (α) −β x delta function of shape parameter order. Also the scale fX (x)dx = β δ (x).e dx (32) 0 0 parameter can be considered as a random variable. Z α Z α = β − .β , (33)

c 2015 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. Lett. 3, No. 2, 61-67 (2015) / www.naturalspublishing.com/Journals.asp 65

which proves our result. And also by using equation (6) with t = 1 and a = 0; we have ∞ α β −tX β −α δ (α) −β x −tx 1 x− −1 (1 − x) −1 E[e ]= (x).e .e dx (34) = Γ (β − α) . dx, 0 Γ α Γ β Z ∞ 0 (− ) ( ) α α β Z = β − δ ( )(x).e−( +t)xdx (35) = Γ (β − α).Γ −1(β − α), 0 α Z α = β − (β +t) . (36) which proves our result. This result has obtained with similar way for −1 < β ≤ 0, 0 < α and 0 < α + β the α α Since the Fourier transform F{δ ( )(x)} = (iw) , which case. implied that the characteristic function (CF) of this α α distribution equals to β − (it + β) , for −1 < α ≤ 0. Therefore, as previous cases, we represented the PDF Therefore, this form of the PDF of Gamma distribution of Beta distribution as the product of fractional derivation indicates that the extended shape parameter appears the of Dirac delta function. Also, we extended the parameter relationship between fractional calculus and Statistics. space from (0,∞) × (0,∞) to (−1,∞) × (0,∞). Also, it allows the representation of the PDF as a product of fractional derivation of Dirac delta function of shape parameter order. Also, the scale parameter can also be considered as a random variable. 6 the Liouville fractional differintegral operator on the MGF of positive random variable, at the origin, is giving fractional 5 The generalized Beta random variable moments nq The Beta distribution is a two-parameter (α and β) family Recently, fractional moments of the type E[X ], where α β N x −1(1 − x) −1 n ∈ and 0 < q ≤ 1, have been introduced [4], such of density functions fX (x) = for quantities have important characteristics: (i) they are B(α,β) exact natural generalization of integer moments as like as 0 ≤ x ≤ 1[7]. It is often used to represent the proportions and percentages. It has the following probability density fractional differential operators generalize the classical function: differential calculus; (ii) the interesting is the relationship between fractional moments and the Γ α β Φ Φ ( + )Dx α+1(x)Dx β +1(y), y = 1 − x (37) fractional special functions. Also, in [5] it was shown that complex fractional moments, which are complex The PDF by using equation (21) with β = 0 and equation moments of order nqth of a certain distribution, are (17), can be written as following definition. equivalent to Caputa fractional derivation of generalized characteristic function in origin, such that when q = 1 the Definition 5.1. suppose that X be a random variable of case was reduced to the complex moments. The fractional Beta distribution as a two-parameter family of distribution moments mentioned above are moments of positive real functions, in which the parameter β and α are the shape order, but in here, they are moments of real order. parameters. Then PDF of this distribution can be defined Now we demonstrate that the fractional moments of a as positive random variable, the αth moments of X, can be obtained directly from Liouville fractional integrals and derivations of the MGF at the origin, as in following Γ α β α β ( + )K+(x)K+(y), theorem. f x Γ β α δ (α) β (38) ( )=  ( − ) (x)K+(y), Γ α β δ (β ) α  ( − ) (y)K+(x), Theorem 6.1. Suppose that X be a positive and continuous random variable and its MGF be infinite. then for 0 ≤ α, where the first expression satisfies for 0 < α and the αth moments of X can be obtained as following: 0 < β. The second expression satisfies for α α −α α −1 < α ≤ 0, 0 < β and 0 < α + β. I+{MX (t)} |t=0= I−{MX (−t)} |t=0= E[X ], > 0 (39) The third expression satisfies for −1 < β ≤ 0, 0 < α and 0 < α + β. and α α α D {M (t)} | = D {M (−t)} | = E[X ], 0 ≤ α ∞ α + X t=0 − X t=0 To show that 0 fX (x)dx = 1 for −1 < ≤ 0, by (40) considering −1 < α ≤ 0, 0 < β and 0 < α + β, we get R in particular when α = n ∈ N, ∞ 1 β −1 (α) (1 − x) fX (x)dx = Γ (β − α) δ (x) dx, n (n) n Γ (β) D {MX (t)} |t 0= M (t) |t 0= E[X ], (41) Z0 Z0 + = X =

c 2015 NSP Natural Sciences Publishing Cor. 66 M. Ganji, F. Gharari: The Generalized Random Variable Appears the Trace...

and for α > 0,X > 0. Also, we have

n n (n) n D−{MX (−t)} |t=0= (−1) MX (−t) |t=0= E[X ]. (42) α d α D {M (−t)} = (−1)m( )mIm− {M (−t)} − X dt − X ∞ d α = (−1)m( )mIm− { e−xt f (x)dx} Proof. We have: dt − X Z0 ∞ d 1 α α −xt = (−1)m( )m I−{MX (−t)} = I−{ e fX (x)dx} Γ α 0 dt (m − ) Z ∞ ∞ ∞ ∞ α 1 α−1 −xu m− −1 −xu = (u −t) e fX (x)dxdu × (u −t) e fX (x)dxdu Γ (α) t 0 Zt Z0 Z Z ∞ ∞ m d m 1 1 α−1 −xu = (−1) ( ) = (u −t) e fX (x)dxdu, Γ α Γ (α) dt (m − ) Zt Z0 ∞ ∞ m−α−1 −xu by using the the Fubini theorem, we get × (u −t) e fX (x)dxdu, Zt Z0 ∞ ∞ 1 α−1 −xu and by using the Fubini theorem, we have = (u −t) e fX (x)dudx, Γ (α) Z0 Zt y d 1 under the substitution u −t = , we have = (−1)m( )m x dt Γ (m − α) ∞ ∞ ∞ ∞ 1 α−1 −α −y −xt m−α−1 −xu = y x e e fX (x)dydx × (u −t) e fX (x)dudx, Γ (α) 0 0 0 t ∞ Z Z Z Z −α −xt y = x e fX (x)dx under the substitution u −t = 0 x Z α = E[X − e−Xt ], d 1 = (−1)m( )m then dt Γ (m − α) ∞ ∞ α −α −Xt −α m−α−1 α−m −y −xt I−{MX (−t)} |t=0= E[X e ] |t=0= E[X ], × y x e e fX (x)dydx 0 0 ∞ Z Z for α > 0,X > 0. With similar way we get α −xt = x e fX (x)dx ∞ 0 α α Z xt α −Xt I+{MX (t)} = I+{ e fX (x)dx} = E[X e ], 0 Z t ∞ 1 α−1 xu then = (t − u) e fX (x)dxdu Γ (α) ∞ Z− Z0 t ∞ 1 α−1 xu = (t − u) e fX (x)dxdu, Γ (α) −∞ 0 α α −Xt α Z Z D−{MX (−t)} |t=0= E[X e ] |t=0= E[X ], again, by using the Fubini theorem, we have for X > 0 and m − 1 ≤ α < m. Also similarly ∞ t 1 α−1 xu = (t − u) e fX (x)dudx, Γ (α) 0 ∞ Z Z− α d m m−α D {MX (t)} = ( ) I {MX (t)} y + dt + ∞ and under the substitution t − u = d α x = ( )mIm− { ext f (x)dx} dt + X ∞ ∞ Z0 1 α−1 −α −y xt d 1 = Γ α y x e e fX (x)dydx m ( ) 0 0 = ( ) Γ α ∞ Z Z dt (m − ) −α xt ∞ ∞ = x e fX (x)dx m−α−1 xu 0 × (t − u) e fX (x)dxdu Z t 0 −α Xt Z Z = E[X e ], d 1 = ( )m dt Γ (m − α) then ∞ ∞ m−α−1 xu α −α Xt −α × (t − u) e fX (x)dxdu, I {MX (t)} |t=0= E[X e ] |t=0= E[X ], + Zt Z0

c 2015 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. Lett. 3, No. 2, 61-67 (2015) / www.naturalspublishing.com/Journals.asp 67

by using the Fubini theorem, we have [4] M. Ganji, N. Eghbali and F. Gharari, Some Fractional Special Functions and Fractional Moments, Gen. Math. Nots. Vol, No.2, October, 120-127, (2013). d 1 [5] M. Ganji and F. Gharari, The Generalized Fourier = ( )m dt Γ (m − α) Transform for the Generation of Complex Fractional ∞ ∞ Moments, International Journal of Mathematical Archive m−α−1 xu × (t − u) e fX (x)dudx, (IJMA),Vol.5,No, 293-299, (2014). Z0 Zt [6] R. Hilfer, Threefold introduction to fractional derivatives, y Wiley-VCH, Weinheim, (2008). under the substitution t − u = [7] V. V. Krishnan, Probability and random processes, John x Wiely and sons (2006). d 1 = ( )m dt Γ (m − α) ∞ ∞ Masoud Ganji is m−α−1 α−m −y xt × y x e e fX (x)dydx Associate Professor at 0 0 ∞ Z Z Department of Statistics, α xt = x e fX (x)dx Faculty of Mathematical 0 Z α Science, University of = E[X eXt ], Mohaghegh Ardabili, Ardabil, Iran. He obtained then his Ph. D. from the National α α Xt α University of Delhi, India. D+{MX (t)} |t=0= E[X e ] |t=0= E[X ], His research interests are in the areas of Applied Mathematics, Statistical inference. He has published for X > 0 and m 1 α < m. − ≤ research articles in reaputed international journals of Statistics and sciences. He is referee and editor of mathematical journals in the frame of Applied Mathematics. He is Director of juornal of Hyerstructures. 7 Perspective Fatemeh Gharari In this paper, we obtained a new relationship between obtained the M.Sc. degree in fractional calculus and statistics by introducing a Mathematical Analysis from generalized random variable. With this definition, the the university of Mohaghegh domain of shape parameter space of distributions such as Ardabili. Her main research Gamma, Weibull and Beta were expanded from (0,∞) to interests are: Distribution (−1,∞). Also, we showed that the Liouville fractional theory, Fractional calculus differintegral operator on the MGF of a positive random and Functional Analysis. variable, at the origin, gives fractional moments.

Acknowledgement

We would like to thank the referees for a careful reading of our paper and lot of valuable suggestions on the first draft of the manuscript.

References

[1] A. Kilbas, M. H. Srivastava and J. J. Trujillo, Theory and application of fractional differential equations, in: North Holland Mathematics Studies, vol.204, (2006). [2] I. Podlubny, Fractional differential equations, San Diego: Academic Press (1999). [3] K. S. Miller and B. Ross, An introduction to fractional calculus and fractional differential equations, John Wiley and sons, (1993).

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