Following from the author’s recent paper at Sampling 2014 which presented a method for calculation of the sampling probability density function due to the particulate heterogeneity (density function of the fundamental sampling uncertainty), it is possible to apply the same characteristic function method to arrive at the overall sampling distribution for any sampling protocol and analysis method. This paper develops the application of the method of characteristic functions to the overall sampling problem including the uncertainty which derives from the primary sampling from a process stream. The assay distribution in a process stream or of impurities in the flow of a final product can be governed by non-Gaussian, serially correlated distributions. The paper shows how such circumstances can be dealt with to arrive at robust solutions. The paper represents an end-point in the theory of sampling as it provides a means of determining the entire distribution function for a sampling system. Such a determination has not previously been possible and having determined the entire distribution function, the statistics of the sampling process are completely determined.