With the availability of fast and high-volume computers at relatively low cost, some of the problems that were considered only of marginal academic interest are gaining prominence in the recent decades. One such problem is the problem of specification of a probability function through its moments. Moments are widely used in scientific disciplines that deal with data in the analysis of statistical distributions. They are applicable to many different aspects of image processing, and are usually discussed in terms of their order. Hence it is common practice in Statistical applications to identify a distribution type by its first few standardized moments. However, the question: ‘How many moments may be good enough to capture the essential features of a distribution’ can prove to be of considerable challenge and interest for further research. Hence, the book is devoted to investigate the problem of generating a class of densities, which have the first k (say, 4) moment’s common but differ in their higher order moments and investigates to what extent and qualitatively in what way; they can differ from one another in an analytical and comprehensive way.