A new probabilistic framework with reliability aspects and statistical analysis of average goals per game in women’s football

The impact of probability distributions is evident across numerous scientific fields, especially within sports sciences. In consideration of the impacted implications of probability distributions, this paper proposes a new probability distribution termed a new trigonometric exponentiated Rayleigh (NTE-Rayleigh) distribution. The NTE-Rayleigh distribution is the outcome of combining the generalized Rayleigh distribution with a notable trigonometric method. In our research, we derive various mathematical properties of the NTE-Rayleigh distribution, with a concentrated focus on its quartile characteristics. A well-established estimation technique is utilized to formulate the mathematical expressions for the estimators of the NTE-Rayleigh distribution. Furthermore, a simulation analysis is undertaken to evaluate the effectiveness of these estimators across different parameter combinations of the NTE-Rayleigh distribution. In closing, an application of the NTE-Rayleigh distribution, informed by insights from sports sciences, is discussed. This presentation aims to illustrate the practical application of the NTE-Rayleigh distribution, with the goal of establishing its superiority over various conventional and non-conventional probability distributions. For this purpose, a number of probability distributions from the literature has been selected. The evaluation based on three key statistical tests, along with the p-value, reveals that the NTE-Rayleigh distribution outperforms all the chosen conventional and non-conventional probability distributions.