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

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The Generalized Random Variable Appears the Trace of Fractional Calculus in Statistics Appl. Math. Inf. Sci. Lett. 3, No. 2, 61-67 (2015) 61 Applied Mathematics & Information Sciences Letters An International Journal http://dx.doi.org/10.12785/amisl/030204 The Generalized Random Variable Appears the Trace of Fractional Calculus in Statistics 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 shape parameter 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 integral of probability density function (PDF), at the origin, for α > 0, and also the fractional derivative of PDF, at the origin, for −1 < α < 0. We also, presented PDFs of such distributions as a product of fractional derivation of Dirac delta function of shape parameter order, α δ ( )(.). Finally, we showed that the Liouville fractional differintegral operator on the moment generating function (MGF) of positive random variable, at the origin, gives fractional moments. Keywords: Fractional calculus, Dirac delta function, Fractional moments, Weibull distribution, Gamma distribution, Beta distribution 1 Introduction Dirac delta function which is defined as: δ C The study of generalized functions is now widely used in : X → applied mathematics and engineering 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 convolution kernels K± ∈ support 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 singular distribution 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... exponential family 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 derivatives and statistics α+1(x) ∼ (1, ( + 1) ), theory can be obtained, this means 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) integrals 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 generalized function 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].
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