A theoretical framework is presented, which derives chemical master equations for the number of protein molecules produced in a given time window, It is applied to derive analytical solutions that describe protein production distributions for the random bursting model (with an exponential or geometric burst-size distribution) and the clustering model. This distribution is experimentally observable using recently developed, single-molecule gene expression experiments. Furthermore, intrinsic stochasticity in a gene's expression can be calculated from protein production distributions using a new, time-dependent noise curve analysis. Different models of gene expression are compared with respect to their protein production distributions and intrinsic stochasticity, revealing the effects of molecular memory and burstlike expression on fluctuations in gene expression. It is distinct from and provides major advantages over measurements of steady-state concentrations.