  Observation of phenotypic diversity in a population of genetically identical
cells is often linked to the stochastic nature of chemical reactions involved
in gene regulatory networks. We investigate the distribution of population
averaged gene expression levels as a function of population, or sample, size
for several stochastic gene expression models to find out to what extent
population averaged quantities reflect the underlying mechanism of gene
expression. We consider three basic gene regulation networks corresponding to
transcription with and without gene state switching and translation. Using
analytical expressions for the probability generating function of observables
and Large Deviation Theory, we calculate the distribution and first two moments
of the population averaged mRNA and protein levels as a function of model
parameters, population size and number of measurements contained in a data set.
We validate our results using stochastic simulations also report exact results
on the asymptotic properties of population averages which show qualitative
differences among different models.
