Transcription in single cells is an inherently stochastic process as mRNA levels vary greatly between cells, even for genetically identical cells under the same experimental and environmental conditions. We present a stochastic two-state switch model for the population of mRNA molecules in single cells where genes stochastically alternate between a more active ON state and a less active OFF state. We prove that the stationary solution of such a model can be written as a mixture of a Poisson and a Poisson-beta probability distribution. This finding facilitates inference for single cell expression data, observed at a single time point, from flow cytometry experiments such as FACS or fluorescence in situ hybridization (FISH) as it allows one to sample directly from the equilibrium distribution of the mRNA population. We hence propose a Bayesian inferential methodology using a pseudo-marginal approach and a recent approximation to integrate over unobserved states associated with measurement error.