Lagrangian expansions can be used to obtain numerous very 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 amount of production for optimum profits. This book is a comprehensive, systematic treatment of the two classes of Lagrangian probability distributions along with some of their sub-families and their properties; important applications are also given.Graduate students and researchers interested in Lagrangian probability distributions, who have sound knowledge of standard statistical techniques, will find this book valuable. It may be used as a reference text or in courses and seminars on distribution theory and Lagrangian distributions. Applied scientists and researchers in environmental statistics, reliability, sales management, epidemiology, operations research, and the optimization of profits in manufacturing and marketing will benefit immensely from the various applications in the book.