Predictive Analysis of Random Determinants Using Chebyshev-Type Inequalities and Polynomial Transformations

The probabilistic behavior of determinants of random matrices has attracted sustained interest due to its relevance in mathematical statistics, reliability theory, and stochastic modeling. While classical determinant theory is well established, the integration of probabilistic inequalities and polynomial approximation techniques into the prediction of random determinants remains an evolving area of research. This article develops a comprehensive analytical framework for predicting random determinants by combining classical results from determinant theory with probabilistic tools such as Chebyshev-type inequalities and polynomial transformations. Drawing upon foundational statistical texts and recent studies on random determinants with independent and identically distributed entries, the study emphasizes distribution-free prediction bounds and analytical tractability. Special attention is given to the role of Chebyshev polynomials and their conversion into power series representations, enabling explicit moment-based approximations of determinant-related random variables. The manuscript synthesizes prior results on Gamma, Weibull, and Bernoulli-distributed matrix elements and extends the discussion toward a unified perspective on variance-driven predictability. By situating recent probabilistic determinant studies within the broader theoretical context of mathematical statistics, the article highlights methodological consistencies, limitations of existing approaches, and opportunities for future extensions. The results contribute to a deeper understanding of how classical inequalities and polynomial methods jointly support robust, non-parametric prediction of random determinants, particularly in situations where exact distributions are analytically intractable.