International Journal of Statistics and Probability; Vol. 8, No. 6; November 2019 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education Partitioning Problems Arising From Independent Shifted-Geometric and Exponential Samples With Unequal Intensities Thierry E. Huillet Correspondence: Thierry E. Huillet, Laboratoire de Physique Th´ eorique et Mod´ elisation Universit´ e de Cergy-Pontoise CNRS UMR-8089 Site de Saint Martin 2 avenue Adolphe-Chauvin 95302 Cergy-Pontoise, France. E-mail: Thierry.Huillet@u-cergy.fr Received: September 2, 2019 Accepted: October 9, 2019 Online Published: October 13, 2019 doi:10.5539/ijsp.v8n6p31 URL: https://doi.org/10.5539/ijsp.v8n6p31 Abstract Two problems dealing with the random skewed splitting of some population into J different types are considered. In a first discrete setup, the sizes of the sub-populations come from independent shifted-geometric with unequal charac- teristics. Various J →∞ asymptotics of the induced occupancies are investigated: the total population size, the number of unfilled types, the index of consecutive filled types, the maximum number of individuals in some state and the index of the type(s) achieving this maximum. Equivalently, this problem is amenable to the classical one of assigning indistinguishable particles (Bosons) at J sites, in some random allocation problem. In a second parallel setup in the continuum, we consider a large population of say J ‘stars’, the intensities of which have independent exponential distributions with unequal inverse temperatures. Stars are being observed only if their intensities exceed some threshold value. Depending on the choice of the inverse temperatures, we investigate the energy partitioning among stars, the total energy emitted by the observed stars, the number of the observable stars and the energy and index of the star emitting the most. Keywords: sum and maximum, independent shifted-geometric/exponential distributions, discrete/continuous partition- ing, combinatorial probability 1. Introduction Consider the partitioning of some population the individuals of which can be of J different types or states. We assume that the sizes of the type- j sub-populations ( j = 1, ..., J ) have independent shifted-geometric distributions with unequal success probabilities. Depending on these probabilities, we envisage various asymptotics for the occupancy distributions, including total population size and the number of unfilled states. Other statistical quantities of interest such as: the index of the consecutive filled states, the maximum number of particles in some state and the index of the site(s) achieving this maximum are also investigated. One of the asymptotics we chiefly focus on is J →∞. A toy variant of the latter model is also investigated in the continuum which is shown to be amenable to a quite similar treatment; it deals with a population of say J ‘stars’ the intensities of which have independent exponential distributions with unequal inverse temperatures that can be observed or not depending on whether the intensities exceed or not some threshold value. Of parallel interest then is the energy partitioning among stars, the total energy emitted by the observed stars, the number of the observable stars, the energy and index of the star emitting the most. And the way all these quantities depend on the choice of the inverse temperatures. Some examples are detailed and the limit J →∞ is also investigated in this context. This second aspect of the partitioning problem in the continuum seems to be new. Let us summarize our results and sketch the organization of the manuscript: motivated by examples from physics, we have studied specific partitioning problems, thereby contributing to general probability theory and discrete mathematics. Con- sidering a population with J different sub-populations (or states) whose sizes G j are independent (shifted-)geometrically distributed, in general with different parameters, we have studied various distributions of interest under several asymptotic regimes. First, in Section 2.1, the distribution of total population size X J = ∑ J j=1 G j , and the joint distributions of relative population sizes G j /E (X J ) were investigated, also asymptotically, the latter e.g. for J fixed and large population size. A condensation phenomenon was highlighted. Section 2.2 considers the number of non-empty states and constrained occupancies problems. In Section 2.3 asymptotics where J →∞ are discussed, with a {0 − 1}-law distinguishing if the series of parameters converges or not. Further, the first non-empty state and site indices till consecutive records (Section 2.4) and the size and index of the most filled state (Sections 2.5 and 2.6) have been addressed. Section 2.7 gives some 31