  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.
