A central challenge of gene expression analysis during the last few decades has been the characterization of the expression patterns experimentally and theoretically. Modern techniques on single-cell and -molecule resolution reveal that transcriptions and translations are stochastic in time and that clonal population of cells displays heterogeneity in the abundance of a given RNA and protein per cell. Hence, to take into account a cell-to-cell variability, we consider a stochastic model of transcription and the chemical master equation. Our stochastic analysis and Monte-Carlo simulation show that the limiting distribution of mRNA copy number can be expressed by a Poisson-beta distribution. The distribution represents the four different types of expression patters, which are typically found in various experimental profiles.