Stratigraphy preserved in alluvial basins holds important information for reconstructing past environmental conditions via inversion methodologies. In this paper we explore, through the use of physical and numerical experiments, the forward problem, that is, we quantify how the probabilistic structure of the processes that govern the evolution of depositional systems relates to the probability distribution of the preserved bed thicknesses. We demonstrate that the extreme variability, as evidenced by heavy-tailed distribution of the surface elevation increments, largely cancels itself out in the resulting stratigraphy. Specifically, we show that bed thickness is well described by an exponential distribution even when erosional and depositional increments characterizing the surface evolution exhibit heavy-tailed statistics, i.e., large, infrequent events have a significant chance of occurrence. We attribute this finding to the symmetric nature of the distribution of elevation increments (both erosional and depositional events) and the additive nature of the stratigraphic filter. We also show that the variability of surface elevation increments, as measured by the interquartile range of their probability distribution, has a robust and well-defined relationship with the preserved mean bed thickness.