CONSTRUCTION OF TRANSMUTED PROBABILITY MASS FUNCTION OF DISCRETE UNIFORM DISTRIBUTION ON A DEFINED INTERVAL [A,B]

Abstract

This study introduces the construction and analysis of a transmuted probability mass function (TPMF) derived from the discrete uniform distribution on a defined interval. The discrete uniform distribution, while widely applied in scenarios where all outcomes are equally likely, is limited by its inability to capture variations in probabilities that arise in practical settings. To address this limitation, the transmutation technique proposed by Shaw and Buckley (2007) is extended to the discrete case, resulting in a more flexible distributional framework. The TPMF is defined by embedding a transmutation parameter, which adjusts the probability structure of outcomes and enhances modeling versatility. The study provides a rigorous construction of the TPMF, beginning with the determination of the normalization constant and establishing its validity through proofs of non-negativity and normalization. Closed-form expressions for key distributional properties such as the mean, variance, and moment generating function (MGF) are derived. Theoretical results are supported with worked examples, demonstrating explicit computation of the PMF, mean, and variance for defined intervals under specified parameter values. These examples verify consistency with probabilistic principles and highlight the adaptability of the TPMF relative to the standard discrete uniform distribution. Findings indicate that the TPMF introduces controlled variability into outcome probabilities without sacrificing analytical tractability. The parameter shifts both expectation and dispersion, making the distribution suitable for contexts where uniformity is unrealistic. Potential applications are discussed in areas such as risk management, decision theory, and fuzzy systems, where probabilistic models must accommodate uncertainty and non-uniform influences. This work thus contributes to probability theory by extending the scope of transmutation methods to discrete distributions, offering a novel tool for statistical modeling and real-life problem solving.