Moment Generating Function, Expectation and Variance of Ubiquitous Distributions with Applications in Decision Sciences: A Review

Statistics have been widely used in many disciplines including science, social science, business, engineering, and many others. One of the most important areas in statistics is to study the properties of distribution functions. To bridge the gap in the literature, this paper presents the theory of some important distribution functions and their moment generating functions. We introduce two approaches to derive the expectations and variances for all the distribution functions being studied in our paper and discuss the advantages and disadvantages of each approach in our paper. In addition, we display the diagrams of the probability mass function, probability density function, and cumulative distribution function for each distribution function being investigated in this paper. Furthermore, we review the applications of the theory discussed and developed in this paper to decision sciences.