In this paper, probabilistic interpretation of the Kober fractional integration of non-integer order is proposed. We prove that the fractional integral, which is proposed by Kober, can be interpreted as an expected value of a random variable up to a constant factor. In this interpretation, the random variable describes dilation (scaling), which has the gamma distribution. The Erdelyi-Kober fractional integration also has a probabilistic interpretation. Fractional differential operators of Kober and Erdelyi-Kober type have analogous probabilistic interpretation. The proposed interpretation leads to a possibility of generalization of the fractional integration and differentiation by using continuous probability distributions.
Digital Object Identifier (DOI)
E. Tarasov, Vasily and S. Tarasova, Svetlana
"Probabilistic Interpretation of Kober Fractional Integral of Non-Integer Order,"
Progress in Fractional Differentiation & Applications: Vol. 5
, Article 1.
Available at: https://dc.naturalspublishing.com/pfda/vol5/iss1/1