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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    characteristic equations
    random variables
    calculus
    Random variables
    probability density functions
    Probability density function
    moments
    Taylor series
    expansion
    Fourier transforms
    Derivatives

    All Science Journal Classification (ASJC) codes

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

    Cita questo

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    title = "On the use of fractional calculus for the probabilistic characterization of random variables",
    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",
    keywords = "Fractional calculus; Generalized Taylor series; Complex order moments; Fractional moments; Characteristic function series; Probability density function series",
    author = "{Di Paola}, Mario and Giulio Cottone",
    year = "2009",
    language = "English",
    volume = "24",
    pages = "321--330",
    journal = "Probabilistic Engineering Mechanics",
    issn = "0266-8920",
    publisher = "Elsevier Limited",

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    TY - JOUR

    T1 - On the use of fractional calculus for the probabilistic characterization of random variables

    AU - Di Paola, Mario

    AU - Cottone, Giulio

    PY - 2009

    Y1 - 2009

    N2 - 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

    AB - 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

    KW - Fractional calculus; Generalized Taylor series; Complex order moments; Fractional moments; Characteristic function series; Probability density function series

    UR - http://hdl.handle.net/10447/41452

    M3 - Article

    VL - 24

    SP - 321

    EP - 330

    JO - Probabilistic Engineering Mechanics

    JF - Probabilistic Engineering Mechanics

    SN - 0266-8920

    ER -