The book deals with bilinear forms in real random vectors and their generalizations as well as zonal polynomials and their applications in handling generalized quadratic and bilinear forms. The book is mostly self-contained. It starts from basic principles and brings the readers to the current research level in these areas. It is developed with detailed proofs and illustrative examples for easy readability and self-study. Several exercises are proposed at the end of the chapters. The complicated topic of zonal polynomials is explained in detail in this book. The book concentrates on the theoretical developments in all the topics covered. Some applications are pointed out but no detailed application to any particular field is attempted. This book can be used as a textbook for a one-semester graduate course on quadratic and bilinear forms and/or on zonal polynomials. It is hoped that this book will be a valuable reference source for graduate students and research workers in the areas of mathematical statistics, quadratic and bilinear forms and their generalizations, zonal polynomials, invariant polynomials and related topics, and will benefit statisticians, mathematicians and other theoretical and applied scientists who use any of the above topics in their areas. Chapter 1 gives the preliminaries needed in later chapters, including some Jacobians of matrix transformations. Chapter 2 is devoted to bilinear forms in Gaussian real ran dom vectors, their properties, and techniques specially developed to deal with bilinear forms where the standard methods for handling quadratic forms become complicated."

In this book we define univariate and bivariate gamma-type distributions and discuss some of their statistical functions, including the moment generating function. Numerous distributions such as the Rayleigh, half-normal and Maxwell distributions can be obtained as special cases. The moment generating function of both univariate and bivariate random variables are derived by making use of the inverse Mellin transform technique and expressed in terms of generalized hypergeometric functions. These representations provide computable expressions for the moment generating functions of several of the distributions that were identified as particular cases. Some other statistical functions are also given in closed form. The univariate distribution is utilized to model two data sets. This model provides a better fit than the two-parameter Weibull model or its shifted counterpart, as measured by the Anderson-Darling and Cramer-von Mises statistics.