Evaluation of the probability distribution of intake from a single measurement on a personal air sampler

An analytical expression has been derived for the k-sum distribution, formed by summing k random variables from a lognormal population. Poisson statistics are used with this distribution to derive the distribution of intake when breathing an atmosphere with a constant particle number concentration. Bayesian inference is then used to calculate the posterior probability distribution of concentrations from a given measurement. This is combined with the above intake distribution to give the probability distribution of intake resulting from a single measurement of activity made by an ideal sampler. It is shown that the probability distribution of intake is very dependent on the prior distribution used in Bayes’ theorem. The usual prior assumption, that all number concentrations are equally probable, leads to an imbalance in the posterior intake distribution. This can be resolved if a new prior proportional to w^_2/3 is used, where w is the expected number of particles collected.