About randomness of aerosol size distributions

All aerosol formation and evolution processes, such as nucleation, condensation, fragmentation, etc., are understood and rationalized via fundamental probabilistic concepts such as probabilities of collision, coagulation, dispersion, etc. Therefore all theoretical size distribution functions (lognormal, modified gamma distribution, self-preserving particle size distribution for Brownian coagulation, etc.) are in fact size probability density functions pdf(r). Any (e.g., measured) size distribution f(r) of an aerosol system is some random realization of its pertinent size probability density function pdf(r). When pdf(r) is viewed as a continuous function, the corresponding size distribution vanishes almost everywhere excluding some randomly set of sizes where f(r)=1. We investigate proximity between f(r) and pdf(r) in finite size intervals and derive expressions for estimation of the standard deviations of several aerosol size-dependent properties arising from randomness of f(r).