This book presents a comprehensive, systematic treatment of the class of Lagrangian probability distributions, along with some of its families, their properties, and important applications. Lagrangian expansions can be used to obtain numerous useful probability models, which have been applied to real life situations including, but not limited to: branching processes, queuing processes, stochastic processes, environmental toxicology, diffusion of information, ecology, strikes in industries, sales of new products, and production targets for optimum profits. Graduate students, applied scientists and researchers in environmental statistics, reliability, sales management, epidemiology, operations research, optimization in manufacturing and marketing, and infectious disease control will benefit immensely from the various applications in the book.