### Abstract

Lingua originale | English |
---|---|

pagine (da-a) | 321-330 |

Numero di pagine | 9 |

Rivista | Probabilistic Engineering Mechanics |

Volume | 24 |

Stato di pubblicazione | Published - 2009 |

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### 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

**On the use of fractional calculus for the probabilistic characterization of random variables.** / Di Paola, Mario; Cottone, Giulio.

Risultato della ricerca: Article

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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 -