The Generalized Random Variable Appears the Trace of Fractional Calculus in Statistics

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1 Appl. Mah. Inf. Sci. Le. 3, No. 2, (215) 61 Applied Mahemaics & Informaion Sciences Leers An Inernaional Journal hp://dx.doi.org/ /amisl/324 The Generalized Rom Variable Appears he Trace of Fracional Calculus in Saisics Masoud Ganji 1, Faemeh Gharari 2 1 Deparmen of Saisics, Universiy of Mohaghegh Ardabili, Ardabil, Iran 2 Deparmen of Mahemaics, Universiy of Mohaghegh Ardabili, Ardabil, Iran Received: 5 Aug. 214, Revised: 14 Mar. 215, Acceped: 17 Mar. 215 Published online: 1 May 215 Absrac: In his paper, inroducing a generalized rom variable, we obained a new relaionship beween fracional calculus saisics. Wih his definiion, he domain of shape parameer space for some of disribuions, such as Gamma, Bea Weibull, were exped from(, ) o( 1, ). I is shown ha he expecaion of such rom variable,φ α (x), coincides wih he fracional inegral of probabiliy densiy funcion (PDF), a he origin, for α >, also he fracional derivaive of PDF, a he origin, for 1<α <. We also, presened PDFs of such disribuions as a produc of fracional derivaion of Dirac dela funcion of shape parameer order, δ (α) (.). Finally, we showed ha he Liouville fracional differinegral operaor on he momen generaing funcion (MGF) of posiive rom variable, a he origin, gives fracional momens. Keywords: Fracional calculus, Dirac dela funcion, Fracional momens, Weibull disribuion, Gamma disribuion, Bea disribuion 1 Inroducion The sudy of generalized funcions is now widely used in applied mahemaics engineering sciences. Generalized funcions are defined as a linear funcional on a space X of convenienly chosen es funcions. For every locally inegrable funcion f Lloc 1 (R), here exiss a disribuion F f : X C defined by: F f (ϕ)= f,ϕ = f(x)ϕ(x)dx (1) where ϕ X is es funcion from a suiable space X of es funcions. A disribuion ha corresponds o funcions via equaion (1) are called regular disribuions. Examples for regular disribuions are he convoluion kernels K± α L 1 loc (R) defined as: K α ± = H(±x)±xα 1 for α > H(x) is a Heaviside uni sep funcion. Disribuions ha are no regular are someimes called singular. An example for a singular disribuion is he (2) Dirac dela funcion which is defined as: δ : X C δ(x)ϕ(x)dx=ϕ() for every es funcion ϕ X. The es funcion space X is usually chosen as a subspace of C (R), he space of infiniely differeniable funcions [6]. In he presen work, we suppose ha X is a posiive rom variable α is he shape parameer of disribuion. The new rom variable is represened as he funcion Φ α (x) defined by Φ α (x) = Xα 1 +. The funcion Φ α (x) can be exended o all complex values of α as a pseudo funcion is a disribuion whose suppor is [, ) excep for he case α =, 1,... Since disribuions or generalized funcions in mahemaical analysis are no really funcions in he classical sense, we also call our proposed rom variable a generalized rom variable. The expecaion of his generalized rom variable coincides wih Riemann-Liouville lef fracional inegral of he PDF, a he origin, for α > Marchaud fracional derivaive of he PDF, a he origin, for 1 < α <. The generalized rom variable appears in some disribuions ha belong o he Corresponding auhor mganji@uma.ac.ir c 215 NSP Naural Sciences Publishing Cor.

2 62 M. Ganji, F. Gharari: The Generalized Rom Variable Appears he Trace... exponenial family like Weibull, Gamma, Bea, Binomial poisson disribuions. By his generalized rom variable, we more exp he domain of shape parameer space. For example, he domain of shape parameer which has been already exped for Gamma Weibull disribuions of α > o α > 1 for Bea disribuion from (, ) (, ) o ( 1, ) (, ). In case of negaive values of he shape parameer space, he relaionship beween fracional derivaives saisics heory can be obained, his means we can wrie he PDF of disribuions like Gamma, Weibull, Bea as a produc of fracional derivaives of Dirac dela funcion. The characerisics of his generalized rom variable are as following: (a) By rewriing Binomial poisson disribuions in erms of he generalized rom variable, hey are respecively ransformed ino Gamma Bea disribuions wih a discree parameer space, ha is [P(x;λ);Φ λ (x)]=[γ(λ ;x,1);φ x (λ)], x=,1,... [Bin(x; p);φ p (x)]=[b(p;x,n x+2);φ x (p)] where x,2,...,n [.,Φ] is he rewriing form of generalized rom variable. (b) The expecaion of his generalized rom variable, Φ α (x), coincides wih Riemann-Liouville lef fracional inegral of he PDF a he origin for α > Marchaud fracional derivaive of he PDF a he origin for 1<α <, ha is, we have where E[Φ α (X)]= (I α f)(), α > (D α f)(), 1<α < (3) (I α f)(x) α 1 f(x+ )d (4) is he Riemann-Liouville lef fracional inegral, while (D α f)(x) α 1 f(x+ ) f(x)}d (5) Γ( α) is he Marchaud fracional derivaive. (c) The ineger fracional derivaives of his generalized rom variable are he generalized rom variable, oo. (d) Anoher propery is Φ α ( a) Φ β ()=Φ α+β ( a), (6) where α > 1 β > 1 such ha α+ β > 1 we used he sar noaion for he convoluion operaion. The proof is easy for α > β >, as insance, see [2]. Oher values of α β can be proved using analyic coninuaion. (e) If X 1,X 2,...,X n be i.i.d. W(x;α,θ) hen Φ α+1 (x) Γ(1,Γ(α+ 1)θ), Φ α+1 (x) Σ n 1 Φ α+1(x i ) B(1,n 1). Also, we demonsrae ha he αh momen, α R, of he posiive rom variable X can be obained direcly from he Liouville fracional derivaion inegraions of he MGF a he origin. 2 Preliminaries In his secion, we inroduce noaions, definiions preliminary facs which are used hroughou his paper. We need some basic definiions properies of he fracional calculus heory he generalized funcions heory which are used furher in his paper. As menioned in references [1] [3], he definiions 1 2 of fracional calculus are as following: Definiion 1. For a funcion f defined on an inerval[a, b], he Riemann-Liouville (R-L) inegrals I α a+ f I α b f of order α C,(R(α)>) are defined, respecively, by (Ia+ α f)() ( ξ) α 1 f(ξ)dξ (7) a (Ib α f)() b (ξ ) α 1 f(ξ)dξ. (8) Also, he lef righ R-L fracional derivaions Da+ α f Db α f of order α C, (R(α) > ) are defined, respecively, by (D α a+ f)()=( d d )n (I n α a+ f)(), (9) (Db α f)()=( d d )n (Ia+ n α f)(). (1) If in above definiion, respecively, a = b =, hen we ge Liouville fracional differinegral. Definiion 2. The Liouville fracional differinegral D α ± is defined by c 215 NSP Naural Sciences Publishing Cor.

3 Appl. Mah. Inf. Sci. Le. 3, No. 2, (215) / 63 D α + f(x)}= f rac1γ( α) x (x ) α 1 f()d, d n dx n D α n + f(x)} (11) where he firs expression saisfies for R(α) < he second expression saisfies for R(α)>;n=[R(α)]+1. Also 1 D α f(x)}= Γ( α) ( 1) n d n dx n x (x ) α 1 f()d, D α n f(x)} (12) wih R(α)< for he firs expression R(α)> such ha n = [R(α)] + 1, for he second expression saisfies. In paricular, when α = n N, hen D + f(x)}= f(x),d n + f(x)}= f (n) (x), (13) D f(x)}= f(x),d n f(x)}=( 1) n f (n) (x). (14) Definiion 3. Le f be a generalized funcion f C (R) wih supp f R +. Then is fracional inegral is he disribuion I+ α f defined as: I α + f,ϕ = Iα f,ϕ = K α + f,ϕ (15) for <α < 1. [6] Now if f C (R) wih supp f R +, hen D+ α f = Dα + (I f)=(k α + K+) f = δ (α) f (2) for all α C. Also, he differeniaion rule holds for all β, α C. I conains D α + Kβ + = K β α + (21) DK β + = Kβ 1 + for all β C as a special case [6]. 3 The generalized Weibull rom variable The Wiebull disribuion was originally used for modeling faigue daa is a presen used exensively in engineering problems for modeling, he disribuion of he lifeime of an objec which consiss of several pars ha fails if any componen par fails [7]. I is imporan o noe ha he fracional calculus has also been applied in describing several PDFs in mahemaical saisics in erms of fracional inegral derivaive of exponenial funcions. For insance, he Wiebull disribuion can be wrien in erms of he fracional derivaive or inegral of e θxα, as for R(α) > [6]. Also, he fracional derivaive of order α wih lower limi is he disribuion D α f(z)}defined as: or θ 1 α Γ(α+ 1)Φ α (x)(d α ;x α e θα )(x), (22) D+ α f,ϕ = Dα f,ϕ = K+ α f,ϕ (16) where α C K+ α (x)= d n dx H(x) xα 1, n[h(x) xα+n 1 R(α)> ], R(α)+n>;n N Γ(α+ n) (17) is he kernel disribuion. For α = one can finds K+ (x) = ( dx d )H(x) = δ(x) D + = I as he ideniy operaor. For he α = n; n N, one can finds K n + (x)=δ (n) (x) (18) where δ (n) is he nh derivaive of he δ disribuion. The kernel disribuion in equaion (17) is given by: K+ α (x)= d dx [H(x) x α Γ(1 α) ]= d dx K1 α + (x) (19) θ 1+α Γ(α+ 1)Φ α (x)(i α ;x α e θα )(x), (23) where he subscrip x α indicaes fracional derivaive or inegral wih respec o he variable x α. In his way represenaion of he PDF migh inroduce novel saisical inerpreaions in he sudy of saisical problems involving such hese PDFs (a leas in some cases). We will define he Weibull disribuion as a wo-parameer family of disribuion funcions, in which he parameer θ > is he scale parameer 1 < α is he shape parameer. So, his disribuion can be defined as Γ 2 (α+ 1)Φ α+1 (θ)dφ α+1 (x)e Γ 2 (α+1)φ α+1 (θ)φ α+1 (x). On he oher h, he scale parameer θ, can also be considered as a rom variable. This means ha, depending on he variable which is being differeniaed, we have he disribuion of ha rom variable. Therefore, he PDF can be wrien as follows: D x e Γ 2 (α+1)φ α+1 (θ)φ α+1 (x). (24) c 215 NSP Naural Sciences Publishing Cor.

4 64 M. Ganji, F. Gharari: The Generalized Rom Variable Appears he Trace... And also, if X be rom variable of Wiebull disribuion as following: Γ 2 (α+ 1)Φ α+1 (θ)dφ α+1 (x)e Γ 2 (α+1)φ α+1 (θ)φ α+1 (x), (25) by using equaion (21) wih β = equaion (17), we can rewrie he PDF in erms of he fracional derivaion of Dirac dela funcion, as following definiion. Definiion 3.1. Suppose ha X be a rom variable of Weibull disribuion as a wo-parameer family of disribuion funcions, in which he parameer θ is he scale parameer α is he shape parameer. Then, PDF of his disribuion can be defined as Γ 2 (α+ 1)Φ f(x)= α+1 (θ)k+(x)e α Γ 2 (α+1)φ α+1 (θ)φ α+1 (x) Γ 2 (1 α)φ 1 α (θ)δ (α) (x)e Γ 2 (1 α)φ 1 α (θ)φ 1 α (x) (26) where he fris expression saisfies for < α he second expression saisfies for 1 < α (For comforable compuaionally, we are showing 1<α wih α.). I can wrie, for <α < 1, f X (x)= Γ 2 (α)φ α (θ)δ (α 1) (x)e Γ 2 (α)φ α (θ)φ α (x). Now we are showing ha f X(x)dx, for he case 1<α, oo. We have f X (x)dx= Γ(1 α)θ α δ (α) (x) by using equaion (19) Γ(1 α)θ α x α e Γ(1 α) dx, = Γ(1 α)θ α δ (α) (x).e Γ(1 α)θ α δ (α 1)(x) dx, under he subsiuion u=δ (α 1) (x), = Γ(1 α)θ α e Γ(1 α)θ αu du, = Γ(1 α)θ α.γ 1 (1 α)θ α, which proves our resul. Also he failure rae (or hazard funcion) for disribuion is given by h(x)= Γ 2 (α+ 1)Φ α+1 (θ)k α + (x), α > Γ 2 (1 α)φ 1 α (θ)δ (α) (x), 1<α. (27) So, we succeed o represen he PDF of Weibull disribuion wih he exended shape parameer space, 1 < α, as a produc of fracional derivaion of Dirac dela funcion of shape parameer order. Also he scale parameer can be considered as a rom variable. 4 The generalized Gamma rom variable In his secion, we will define he Gamma disribuion as a wo-parameer family of disribuion funcions, in which he parameer β > is he scale parameer, 1 < α is he shape parameer. This disribuion is defined as Γ(α+ 1)Φ α+1 (β)dφ α+1 (x)e Φ 2(β)Φ 2 (x), The scale parameer β, can also be considered as a rom variable. This means ha, depending on which variable is under differeniaion, we have he disribuion of ha rom variable; ha is, he PDF can be rewrien as: Γ(α+ 1)Φ α+1 (β)d x Φ α+1 (x)exp Φ 2 (β)φ 2 (x), (28) or as Γ(α+ 1)D β Φ α+1 (β)φ α+1 (x)e Φ 2(β)Φ 2 (x). (29) Similarly as a general case of Weibull disribuion, if X be a rom variable, by using equaion (21) wih β = equaion (17), we can wrie is PDF as following definiion: Definiion 4.1. Suppose ha X be a rom variable of Gamma disribuion as a wo-parameer family of disribuion funcions, in which he parameer β is he scale parameer α is he shape parameer. Then, PDF of his disribuion can be defined as f X (x)= Γ(α+ 1)Φ α+1 (β)k α + (x)e Φ 2(β)K 2 + (x) Γ(1 α)φ 1 α (β)δ (α) (x)e Φ 2(β)K 2 + (x) (3) where he firs expression saisfies for < α he second expression saisfies for 1 < α. I can wrie, for <α < 1, f X (x)= Φ α (β)δ (α 1) (x)e Φ 2(β)K 2 + (x). The MGF of he Gamma generalized rom variable is as following: M X ( )=β α (+ β) α, 1<α. (31) Since Lδ (α) (x)} = s α, which implied ha f X(x)dx for 1 < α he MGF be as above, such ha we have f X (x)dx=β α δ (α) (x).e β x dx (32) = β α.β α, (33) c 215 NSP Naural Sciences Publishing Cor.

5 Appl. Mah. Inf. Sci. Le. 3, No. 2, (215) / 65 which proves our resul. And also E[e X ]=β α δ (α) (x).e β x.e x dx (34) = β α δ (α) (x).e (β+)x dx (35) = β α (β + ) α. (36) Since he Fourier ransform Fδ (α) (x)} = (iw) α, which implied ha he characerisic funcion (CF) of his disribuion equals o β α (i+ β) α, for 1<α. Therefore, his form of he PDF of Gamma disribuion indicaes ha he exended shape parameer appears he relaionship beween fracional calculus Saisics. Also, i allows he represenaion of he PDF as a produc of fracional derivaion of Dirac dela funcion of shape parameer order. Also, he scale parameer can also be considered as a rom variable. 5 The generalized Bea rom variable The Bea disribuion is a wo-parameer (α β ) family of densiy funcions f X (x) = xα 1 (1 x) β 1 for B(α,β) x 1 [7]. I is ofen used o represen he proporions percenages. I has he following probabiliy densiy funcion: Γ(α+ β)d x Φ α+1 (x)d x Φ β+1 (y), y=1 x (37) The PDF by using equaion (21) wih β = equaion (17), can be wrien as following definiion. Definiion 5.1. suppose ha X be a rom variable of Bea disribuion as a wo-parameer family of disribuion funcions, in which he parameer β α are he shape parameers. Then PDF of his disribuion can be defined as Γ(α+ β)k+ α (x)kβ +(y), f(x)= Γ(β α)δ (α) (x)k+(y), β (38) Γ(α β)δ (β) (y)k+(x), α where he firs expression saisfies for < α < β. The second expression saisfies for 1<α, <β <α+ β. The hird expression saisfies for 1< β, < α <α+ β. To show ha f X(x)dx for 1 < α, by considering 1<α, <β <α+ β, we ge 1 f X (x)dx= Γ(β α) δ (α) (x) (1 x)β 1 dx, Γ(β) by using equaion (6) wih a = ; we have 1 = Γ(β α) = Γ(β α).γ 1 (β α), x α 1.(1 x)β 1 Γ( α) Γ(β) dx, which proves our resul. This resul has obained wih similar way for 1 < β, < α < α+ β he case. Therefore, as previous cases, we represened he PDF of Bea disribuion as he produc of fracional derivaion of Dirac dela funcion. Also, we exended he parameer space from(, )(, ) o( 1, )(, ). 6 he Liouville fracional differinegral operaor on he MGF of posiive rom variable, a he origin, is giving fracional momens Recenly, fracional momens of he ype E[X nq ], where n N < q 1, have been inroduced [4], such quaniies have imporan characerisics: (i) hey are exac naural generalizaion of ineger momens as like as fracional differenial operaors generalize he classical differenial calculus; (ii) he ineresing poin is he relaionship beween fracional momens he fracional special funcions. Also, in [5] i was shown ha complex fracional momens, which are complex momens of order nqh of a cerain disribuion, are equivalen o Capua fracional derivaion of generalized characerisic funcion in origin, such ha when q he case was reduced o he complex momens. The fracional momens menioned above are momens of posiive real order, bu in here, hey are momens of real order. Now we demonsrae ha he fracional momens of a posiive rom variable, he αh momens of X, can be obained direcly from Liouville fracional inegrals derivaions of he MGF a he origin, as in following heorem. Theorem 6.1. Suppose ha X be a posiive coninuous rom variable is MGF be infinie. hen for α, he αh momens of X can be obained as following: I α +M X ()} = = I α M X ( )} = = E[X α ], α > (39) D α + M X()} = = D α M X( )} = = E[X α ], α (4) in paricular when α = n N, D n +M X ()} = = M (n) X () == E[X n ], (41) c 215 NSP Naural Sciences Publishing Cor.

6 66 M. Ganji, F. Gharari: The Generalized Rom Variable Appears he Trace... D n M X( )} = =( 1) n M (n) X ( ) == E[X n ]. (42) Proof. We have: I α M X( )}=I α e x f X (x)dx} (u ) α 1 e xu f X (x)dxdu by using he he Fubini heorem, we ge under he subsiuion u = y, we have x hen = x α e x f X (x)dx = E[X α e X ], (u ) α 1 e xu f X (x)dxdu, (u ) α 1 e xu f X (x)dudx, y α 1 x α e y e x f X (x)dydx I α M X ( )} = = E[X α e X ] = = E[X α ], for α >,X >. Wih similar way we ge I+ α M X()}=I+ α e x f X (x)dx} ( u) α 1 e xu f X (x)dxdu ( u) α 1 e xu f X (x)dxdu, again, by using he Fubini heorem, we have under he subsiuion u= y x hen = x α e x f X (x)dx = E[X α e X ], ( u) α 1 e xu f X (x)dudx, y α 1 x α e y e x f X (x)dydx I α +M X ()} = = E[X α e X ] = = E[X α ], for α >,X >. Also, we have D α M X( )}=( 1) m ( d d )m I m α M X ( )} =( 1) m ( d d )m I m α e x f X (x)dx} d )m Γ(m α) (u ) m α 1 e xu f X (x)dxdu d )m Γ(m α) (u ) m α 1 e xu f X (x)dxdu, by using he Fubini heorem, we have d )m Γ(m α) under he subsiuion u = y x d )m Γ(m α) (u ) m α 1 e xu f X (x)dudx, y m α 1 x α m e y e x f X (x)dydx = x α e x f X (x)dx = E[X α e X ], hen D α M X( )} = = E[X α e X ] = = E[X α ], for X > m 1 α < m. Also similarly D+M α X ()}=( d d )m I+ m α M X ()} =( d d )m I m α + d )m Γ(m α) d )m Γ(m α) e x f X (x)dx} ( u) m α 1 e xu f X (x)dxdu ( u) m α 1 e xu f X (x)dxdu, c 215 NSP Naural Sciences Publishing Cor.

7 Appl. Mah. Inf. Sci. Le. 3, No. 2, (215) / 67 by using he Fubini heorem, we have d )m Γ(m α) under he subsiuion u= y x d )m Γ(m α) = x α e x f X (x)dx = E[X α e X ], hen ( u) m α 1 e xu f X (x)dudx, y m α 1 x α m e y e x f X (x)dydx D α + M X()} = = E[X α e X ] = = E[X α ], for X > m 1 α < m. 7 Perspecive In his paper, we obained a new relaionship beween fracional calculus saisics by inroducing a generalized rom variable. Wih his definiion, he domain of shape parameer space of disribuions such as Gamma, Weibull Bea were exped from (, ) o ( 1, ). Also, we showed ha he Liouville fracional differinegral operaor on he MGF of a posiive rom variable, a he origin, gives fracional momens. [4] M. Ganji, N. Eghbali F. Gharari, Some Fracional Special Funcions Fracional Momens, Gen. Mah. Nos. Vol, No.2, Ocober, , (213). [5] M. Ganji F. Gharari, The Generalized Fourier Transform for he Generaion of Complex Fracional Momens, Inernaional Journal of Mahemaical Archive (IJMA),Vol.5,No, , (214). [6] R. Hilfer, Threefold inroducion o fracional derivaives, Wiley-VCH, Weinheim, (28). [7] V. V. Krishnan, Probabiliy rom processes, John Wiely sons (26). Masoud Ganji is Associae Professor a Deparmen of Saisics, Faculy of Mahemaical Science, Universiy of Mohaghegh Ardabili, Ardabil, Iran. He obained his Ph. D. from he Naional Universiy of Delhi, India. His research ineress are in he areas of Applied Mahemaics, Saisical inference. He has published research aricles in reapued inernaional journals of Saisics sciences. He is referee edior of mahemaical journals in he frame of Applied Mahemaics. He is Direcor of juornal of Hyersrucures. Faemeh Gharari obained he M.Sc. degree in Mahemaical Analysis from he universiy of Mohaghegh Ardabili. Her main research ineress are: Disribuion heory, Fracional calculus Funcional Analysis. Acknowledgemen We would like o hank he referees for a careful reading of our paper lo of valuable suggesions on he firs draf of he manuscrip. References [1] A. Kilbas, M. H. Srivasava J. J. Trujillo, Theory applicaion of fracional differenial equaions, in: Norh Holl Mahemaics Sudies, vol.24, (26). [2] I. Podlubny, Fracional differenial equaions, San Diego: Academic Press (1999). [3] K. S. Miller B. Ross, An inroducion o fracional calculus fracional differenial equaions, John Wiley sons, (1993). c 215 NSP Naural Sciences Publishing Cor.