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Several theoretical analyses of the dynamics of particle swarms have been offered in the literature over the last decade. Virtually all rely on substantial simplifications, often including the assumption that the particles are deterministic. This has prevented the exact characterization of the sampling distribution of the particle swarm optimizer (PSO). In this paper we introduce a novel method that allows us to exactly determine all the characteristics of a PSO sampling distribution and explain how it changes over any number of generations, in the presence stochasticity. The only assumption we make is stagnation, i.e., we study the sampling distribution produced by particles in search for a better personal best. We apply the analysis to the PSO with inertia weight, but the analysis is also valid for the PSO with constriction and other forms of PSO.