Measurement of airborne radon daughters : a Bayesian approach

The standard mathematical treatment of the build-up and decay of airborne radionuclides on a filter paper uses the solutions of the so-called Bateman equations adapted to the sampling process. These equations can be interpreted as differential equations for the expectation of an underlying stochastic process, which describes the random fluctuations in the accumulation and decay of the sampled radioactive atoms. The probability distribution for the number of ²¹⁸Po, ²¹⁴Pb and ²¹⁴Bi atoms, accumulated after sampling time t, is the product of three Poisson distributions. We show that the distribution of the number of counts, registered by a detector with efficiency &epsilon during a counting period T after the end of sampling, is also the product of three Poisson distributions. Its mean is dependent on ε, t, T, flowrate, and N^0A, N^0B and N^0C, the number of ²¹⁸Po, ²¹⁴Pb and ²¹⁴Bi atoms per unit volume. This joint Poisson distribution was used to construct the likelihood given the observed number of counts. Using Bayes' Theorem we obtained posterior densities for N^0A, N^0B and N^0C. These densities characterize the remaining uncertainty about the unknown airborne concentrations of ²¹⁸Po, ²¹⁴Pb and ²¹⁴Bi atoms.