This study introduces a new continuous probability distribution, termed the extended bounded sine hyperbolic (EBSH) distribution, for modeling non-negative data with flexible structural properties. The distribution accommodates symmetric and skewed behaviors and captures a wide range of hazard rate patterns, including increasing, decreasing, and bathtub-shaped forms. Beyond its theoretical contribution, the study investigates the use of the EBSH distribution as a feature engineering mechanism in machine learning. Raw input variables are transformed through the EBSH formulation to enhance data representation and improve predictive performance. The approach is evaluated using COVID-19 mortality and breast cancer datasets, using models such as recurrent neural networks (RNN) and support vector regression (SVR). Experimental results indicate that EBSH-based feature engineering improves prediction accuracy compared to raw features. For the COVID-19 dataset, the RNN model achieved a mean absolute error (MAE) of 0.0296 and a root mean square error (RMSE) of 0.0373 using raw features, which further reduced to approximately 0.0251 (MAE) and 0.0328 (RMSE) after applying EBSH-based transformations. Similarly, for the breast cancer dataset, SVR produced an MAE of 4.0177 and an RMSE of 4.9926 with raw features, improving to about 3.6842 and 4.5213, respectively, under the engineered feature space. These findings demonstrate that the proposed EBSH distribution not only provides a flexible statistical modeling approach but also serves as an effective feature engineering tool that enhances machine learning performance across real-world datasets.