The $I$-function distribution and its extensions

In this paper we introduce a new probability distribution on $(0,\infty)$ associated with the $I$-function, and hence called the $I$-function distribution. This distribution generalizes several known distributions with positive support (see the table at the end of the paper). It is also shown that the product, quotient, and rational power of independent random variates with $I$-distribution are random variates with $I$-distribution. Another new distribution—the $I$-function Gaussian distribution ($IFIG$ distribution)—is introduced and defined in terms of the $I$-function. For this distribution, the representations of its Mellin and Laplace transforms are obtained. The utilities of the $I$-function distribution are discussed with an application to the likelihood ratio statistic.