  The essentials of fractional calculus according to different approaches that
can be useful for our applications in the theory of probability and stochastic
processes are established. In addition to this, from this fractional integral
one can list out almost all the extended densities for the pathway parameter $q
< 1$ and $q \rightarrow 1$. Here we bring out the idea of thicker or thinner
tailed models associated with a gamma type distribution as a limiting case of
pathway operator. Applications of this extended gamma model in Statistical
Mechanics, input-output models, and solar spectral irradiance modeling etc are
established.
