The primary objective of the work presented in this paper is to introduce and explore a new probability distribution function that incorporates incomplete \(\aleph \) -functions. These functions play a crucial role in the formulation of this distribution, allowing for a broader range of applications in statistical modeling. In addition to introducing the new distribution, the paper discusses several essential mathematical concepts that underpin its formulation. Among these, the Mellin and Laplace transforms are highlighted, as they provide significant tools for analyzing the properties of the newly defined distribution. These transforms facilitate the derivation of various statistical measures and characteristics, which are crucial for understanding the behavior of the distribution. Furthermore, the paper delves into the properties of the inverse Gaussian distribution as it relates to the incomplete \(\aleph \) -function. This examination not only illustrates the connection between the new distribution and existing statistical models but also showcases the potential applications of these functions in probability theory and related fields. Overall, this work aims to enrich the study of probability distributions by integrating incomplete \(\aleph \) -functions, thereby offering new avenues for research and practical applications in statistical analysis.