  The study of probability distributions for random variables and their
algebraic combinations has been a central focus driving the advancement of
probability and statistics. Since the 1920s, the challenge of calculating the
probability distributions of sums, differences, products, and quotients of
independent random variables have drawn the attention of numerous statisticians
and mathematicians who studied the algebraic properties and relationships of
random variables. Statistical distributions are highly helpful in data science
and machine learning, as they provide a range of possible values for the
variables, aiding in the development of a deeper understanding of the
underlying problem. In this paper, we have presented a new probability
distribution based on the $\hat{I}$-function. Also, we have discussed the
applications of the $\hat{I}$ function, particularly in deriving the
distributions of product and the quotient involving two independent $\hat{I}$
function variates. Additionally, it has been shown that both the product and
quotient of two independent $\hat{I}$-function variates also follow the
$\hat{I}$-function distribution. Furthermore, the new distribution, known as
the $\hat{I}$-function distribution, includes several well-known classical
distributions such as the gamma, beta, exponential, normal H-function, and
G-function distributions, among others, as special cases. Therefore, the
$\hat{I}$-function distribution can be considered a characterization or
generalization of the above-mentioned distributions.
