This monograph is, as far as the author has gathered, the third one of its kind which presents various characterisations of many important continuous distributions. It consists of two chapters. The first chapter lists cumulative distributions and probability density functions of six hundred and sixty-seven newly proposed univariate continuous distributions. Chapter Two consists of four sections. Section 2.1 provides characterisations of the majority of the distributions mentioned in Chapter One, based on the ratio of two truncated moments. Section 2.2 takes up the characterizations of some of these distributions in terms of their hazard functions. Section 2.3 deals with the characterizations some of these distributions based on their reverse hazard functions. Characterizations of some of these distributions based on the conditional expectations of certain functions of the random variable are presented in Section 2.4. As pointed out in our previous Monographs (I & II), a good number of proposed distributions in this volume have already been introduced in the literature.

This monograph is, as far as the author has gathered, the second of its kind (the first one was published by Nova in 2017 with coauthors Hamedani and Maadooliat) which presents various characterizations of a wide variety of continuous distributions. These two monographs could also be used as sources to prevent reinventing and duplicating the already exiting distributions. The current book consists of seven chapters. The first chapter lists cumulative and density functions of two hundred and twenty univariate distributions. Chapter two provides characterizations of these distributions: (i) based on the ration of two truncated moments; (ii) in terms of the hazard function; (iii) in terms of the reverse hazard function; (iv) based on the conditional expectation of certain functions of the random variable. Chapter three includes the characterizations of twenty distributions, which appeared in a published paper (Hamedani and Safavimanesh, 2017). Chapter four presents characterizations of thirty six distributions, contains a published paper (Hamedani, 2017). Chapter five covers the characterizations of forty one distributions, which appeared in a published paper (Hamedani, 2018a). Chapter six presents characterizations of eighty distributions, contained in a published paper (Hamedani, 2018b). Finally, chapter seven consists of seventy proposed distributions. The main reason to include previously published papers in Chapters 3-6 is to provide a rather complete source for the interested researchers who would want to avoid reinventing the existing distributions.

The concept of sub-independence is defined in terms of the convolution of the distributions of random variables, providing a stronger sense of dissociation between random variables than that of uncorrelatedness. If statistical tests reject independence but not lack of correlation, a model with sub-independent components can be appropriate to determine the distribution of the sum of the random variables. This monograph presents most of the important classical results in probability and statistics based on the concept of sub-independence. This concept is much weaker than that of independence and yet can replace independence in most limit theorems as well as well-known results in probability and statistics. This monograph, the first of its kind on the concept of sub-independence, should appeal to researchers in applied sciences where the lack of independence of the uncorrelated random variables may be apparent but the distribution of their sum may not be tractable.

The exponential distribution is often used to model the failure time of manufactured items in production. If X denotes the time to failure of a light bulb of a particular make, with exponential distribution, then P(X>x) represent the survival of the light bulb. The larger the average rate of failure, the bigger will be the failure time. One of the most important properties of the exponential distribution is the memory-less property. This book presents various properties of the exponential distribution and inferences about them.