A new probability distribution called the $I$-function distribution has been introduced in this paper, which is related to the $I$-function and generalizes several known distributions. It has been shown that multiplying, dividing, or raising to a power independent random numbers with $I$-distribution results in random numbers with $I$-distribution. Additionally, a new distribution called the $I$-function Gaussian distribution ($IFIG$ distribution) has been defined using the $I$-function, and its Mellin and Laplace transforms have been determined. The $I$-function distribution has practical applications, including in the likelihood ratio statistic.