## 1 Introduction
According to [PERSON], _\"There are no areas that are more vulnerable to such misfortune than sampling and assaying for trace amounts of constituents of interest in the environment, in high purity materials, in precious metals exploration, food chain, chemicals, and pharmaceutical products\"_[15]. Therefore, the accurate evaluation of the concentration value of the targeted analyte at the lot scale and of its uncertainty has two benefits: (1) triggering investment for CRM harvesting from waste streams and (2) preventing decision risk in the mining for precursors of CRMs or precious metals. Additionally, the application of a conservative approach to conformity assessment can mitigate erroneous decisions due to false positives and false negatives when trace element contents are tested [16; 17]. Statistical data analysis in the sampling of grained/particulate solid wastes for trace constituents is often based on mass/discrete probability distributions such as Poisson distribution, double Poisson distribution, or binomial distribution [15; 18; 19; 20; 21; 22]. In the frame of discrete distributions, a granular waste lot is conceptually modeled as a population of particles, with each one having (or not) a targeted property such as a chemical concentration, a nugget mineral, or a generic property. In this context, sampling from a population of discrete materials (e.g., a solid waste lot) is the random selection of a subset (i.e., a \"sample\"). The sample/subset is representative if it contains reliably similar proportions between particles of interest and the particles occurring in the batch of materials to be characterized. In the case of tailings and other fine-grained substances, this approach has at least three drawbacks:  
(i) The concentration of an analyte in a waste particle is a rather continuous function. This contradicts the hypothesis that some particles contain a constant quantity of analyte while others contain none. Accordingly, discrete distributions are unsuitable for the interpretation of statistical data in cases of fine-grained solid wastes. (ii) An analyte can exist in different particles as part of dissimilar compounds. This contradicts the assumption that a particle contains a constant quantity of analyte, further rendering discrete probability models inappropriate. (iii) Discrete sampling (particle by particle) is inappropriate in the case of fine-grained wastes such as tailings and bottom ashes. Instead, incremental sampling is the standard approach used when sampling fine-grained waste dumps [14]. The analytical procedures used for trace elements require sample comminution, i.e., crushing, milling, and sometimes analyte extraction through chemical or physical methods [14; 23; 24]. A well-prepared sample results in a homogenized analyte distribution, meaning that the analytical results reflect the mean concentration of the aliquot. However, even with homogenization, the dependence of analytical results on aliquot size cannot be eliminated. Our proposed approach aims to eliminate the unproven hypothesis of the granularity of the analyte (nudged distribution) contained in the targeted lot, which was often adopted in previous studies without supporting evidence [15; 20]. To overcome the limitations of conventional approaches, we present a new theoretical model designed to enhance data interpretation for the incremental sampling of fine-grained substances, including tailings, pharmaceutical powders, and food [24]. The key concept of the model is assigning a uniform probability density function **(pdf)** to the analyte mass in an aliquot and a triangular **pdf** to the aliquot mass. The rationale for assigning a uniform **pdf** to the mass of the targeted analyte is based on the evidence that 1 g of fine-grained subsample may contain over a million particles of mixed minerals. Further, an analyte may exist as grains or in an atomic state incorporated into different minerals. ISO 98-3 recommends using a uniform **pdf** when little information is available about an assessed quantity [25].