Motivated by real data sets, that does not fit one of the parametric distributions, many well-known non-parametric families of life distributions have been defined and all their properties are explored. In particular their convolution (adding life of independent components), closure formation of systems of n independent components (parallel systems, series system, k-out-of n systems), and mixtures of these life lengths conditioned on different processing (working) environments, are established. However, the researcher, and after surveying the available non-parametric families, has realized that there is still many gaps on handling real data sets, for which new families have to be introduced and studied.From another point of view, all the introduced families of distributions had the assumption that the variables under consideration are continuous. However, in reality, life is measured in days, months, years, weeks or even hours (i.e., integral values) so, the assumption of continuity does not hold.One aim of the research is to define discrete analogs to the continuous families. Once a discrete family is introduced; its properties have to be established.
