On the use of fractional calculus for the probabilistic characterization of random variables

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Abstract

In this paper, the classical problem of the probabilistic characterization of a random variable is reexamined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of α-stable random variables.Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are obtained. Firstly, it is shown that the fractional derivatives of the CF in zero coincide with fractional moments. This is true also in case of CF not derivable in zero (like the CF of α-stable r.vs). Moreover, it is shown that the CF may be represented by a generalized Taylor expansion involving fractional moments. The generalized Taylor series proposed is also able to represent the PDF in a perfect dual representation to that in terms of CF. The PDF representation in terms of fractional
Lingua originaleEnglish
pagine (da-a)321-330
Numero di pagine9
RivistaProbabilistic Engineering Mechanics
Volume24
Stato di pubblicazionePublished - 2009

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All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Civil and Structural Engineering
  • Nuclear Energy and Engineering
  • Condensed Matter Physics
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering

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