Characterization of Some Probability Distributions Based on Conditional Expectation and Variance

The characterizations of probability distributions have been studied by several academics. When a certain distribution is the only one that correlates with a given property, a characterization theorem in probability and statistics is applicable. Additionally, a characterization is a particular distributional or statistical property of a statistic or statistics that describes the corresponding stochastic model in a unique way. Some scholars proposed that real-world data should be characterized under certain criteria before a probability distribution is applied to it. This research investigates the characterization of specific probability distributions using advanced statistical methods. Characterization theorems are fundamental in probability and statistics, as they establish the unique properties that define a given distribution. The study focuses on a novel approach to characterizing truncated negative binomial and logarithmic series distributions through conditional expectation and variance functions. The necessary and sufficient conditions are derived for these characterizations, with a particular emphasis on the failure rate as a key parameter. The findings provide a robust framework for identifying underlying distributions based on their failure rates and conditional properties. An illustrative example is presented to demonstrate the practical application of this methodology, showcasing its potential for uncovering the properties of unobserved distributions.