G G Hamedani

This work is a continuation of our previous works (Hamedani and Safavimanesh, 2017), (Hamedani 2017), (Hamedani and Maadooliat, 2017), (Hamedani, 2018a, 2018b), (Hamedani, 2019), (Hamedani, 2021) and (Hamedani, 2023) on characterizations of distributions introduced in 2021-2023. The current work, our previous works mentioned above may serve as a source of preventing the reinvention and/or duplication of the existing distributions in future. As we pointed out in our papers and monographs, a good number of proposed distributions have already been introduced in the literature. We believe the authors should do a detailed literature search before devoting considerable time on the already existing distributions.

As we mentioned in our previous works, sometimes in real life cases, it is very difficult to obtain samples from a continuous distribution. The observed values are generally discrete due to the fact that they are not measured in continuum. In some cases, it may be possible to measure the observations via a continuous scale, however, they may be recorded in a manner in which a discrete model seems more suitable. Consequently, the discrete models are appearing quite frequently in applied fields and have attracted the attention of many researchers. Characterizations of distributions are important to many researchers in the applied fields. An investigator will be vitally interested to know if their model fits the requirements of a particular distribution. To this end, one will depend on the characterizations of this distribution which provide conditions under which the underlying distribution is indeed that particular distribution.

As we mentioned in our previous works, sometimes in real life cases, it is very difficult to obtain samples from a continuous distribution. The observed values are generally discrete due to the fact that they are not measured in continuum. In some cases, it may be possible to measure the observations via a continuous scale, however, they may be recorded in a manner in which a discrete model seems more suitable. Consequently, the discrete models are appearing quite frequently in applied fields and have attracted the attention of many researchers.Characterizations of distributions are important to many researchers in the applied fields. An investigator will be vitally interested to know if their model fits the requirements of a particular distribution. To this end, one will depend on the characterizations of this distribution which provide conditions under which the underlying distribution is indeed that particular distribution.

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.

Sometimes in real life cases, it is very difficult to obtain samples from a continuous distribution. The observed values are generally discrete due to the fact that they are not measured in continuum. In some cases, it may be possible to measure the observations via a continuous scale, however, they may be recorded in a manner in which a discrete model seems more suitable. Consequently, the discrete models are appearing quite frequently in applied Fields and have attracted the attention of many researchers in the applied fields. An investigator will be vitally interested to know if their model fits the requirements of a particular distribution. To this end, one will depend on the characterizations of this distribution which provide conditions under which the underlying distribution is indeed that particular distribution. This monograph is the continuation of our (2016) monograph which presents further characterizations of many recently introduced important discrete distributions.

The exponential distribution is often used to model the failure time of manufactured items in production. This book presents various properties of the exponential distribution and inferences about them.