Progress in Fractional Differentiation & Applications

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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.

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