6533b82ffe1ef96bd129489e
RESEARCH PRODUCT
On the use of fractional calculus for the probabilistic characterization of random variables
Mario Di PaolaGiulio Cottonesubject
Characteristic function (probability theory)FOS: Physical sciencesAerospace EngineeringMathematics - Statistics TheoryOcean EngineeringProbability density functionComplex order momentStatistics Theory (math.ST)Fractional calculusymbols.namesakeIngenieurwissenschaftenFOS: MathematicsTaylor seriesApplied mathematicsCharacteristic function serieMathematical PhysicsCivil and Structural EngineeringMathematicsGeneralized Taylor serieMechanical EngineeringStatistical and Nonlinear PhysicsProbability and statisticsMathematical Physics (math-ph)Condensed Matter PhysicsFractional calculusFourier transformNuclear Energy and EngineeringPhysics - Data Analysis Statistics and ProbabilitysymbolsFractional calculus; Generalized Taylor series; Complex order moments; Fractional moments; Characteristic function series; Probability density function seriesddc:620Series expansionFractional momentProbability density function seriesSettore ICAR/08 - Scienza Delle CostruzioniRandom variableData Analysis Statistics and Probability (physics.data-an)description
In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. 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 $\alpha$--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 $\alpha$--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 moments is especially accurate in the tails and this is very important in engineering problems, like estimating structural safety.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2009-07-01 |