A stochastic model for a superposition of uncorrelated pulses with a random distribution of amplitudes, sizes, and velocities is presented. The pulses are assumed to move radially with fixed shape and amplitudes decaying exponentially in time due to linear damping. The pulse velocities are taken to be time-independent but randomly distributed. The implications of a distribution of pulse amplitudes, sizes, and velocities are investigated. Closed-form expressions for the cumulants and probability density functions for the process are derived in the case of exponential pulses and a discrete uniform distribution of pulse velocities. The results describe many features of the boundary region of magnetically confined plasmas, such as high average particle densities, broad and flat radial profiles, and intermittent large-amplitude fluctuations. The stochastic model elucidates how these phenomena are related to the statistical properties of blob-like structures. In particular, the presence of fast pulses generally leads to flattened far scrape-off layer profiles and enhanced intermittency, which amplifies plasma–wall interactions.