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It is shown that a Mittag-Leffler density has interesting properties. The Mittag-Leffler random variable has a structural representation in terms of a positive Lévy variable and a power of a gamma variable where these two variables are independently distributed. It is shown that several central limit type properties hold but the limiting forms are positive Lévy variable rather than a Gaussian variable. A path is constructed from a Mittag-Leffler function to the Mathai pathway model which also provides paths to nonextensive statistics and superstatistics.

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In this article we examine the densities of a product and a ratio of two real positive scalar random variables x1 and x2, which are statistically independently distributed, and we consider the density of the product u1 = x1x2 as well as the density of the ratio u2 = and show that Kober operator of the second kind is available as the density of u1 and Kober operator of the first kind is available as the density of u2 when x1 has a type-1 beta density and x2 has an arbitrary density. We also give interpretations of Kober operators of the second and first kind as Mellin convolution for a product and ratio respectively. Then we look at various types of generalizations of the idea thereby obtaining a large collection of operators which can all be called generalized Kober operators. One of the generalizations considered is the pathway idea where one can move from one family of operators to another family and yet another family and eventually end up with an exponential form. Common generalizations in terms of a Gauss' hypergeometric series is also given a statistical interpretation and put on a more general structure so that the standard generalizations given by various authors, including Saigo operators, are given statistical interpretations and are derivable as special cases of the general structure considered in this article.

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