Transcription is the process by which messenger RNA (mRNA) transcripts are synthesized from genes. Measurements in individual living cells reveal fluctuations in mRNA copy numbers over time suggesting that transcription is an intrinsically random process driven by thermal molecular motion of biochemical species. We here use a stochastic model of the transcription process that captures both the extent and timescale of fluctuations in mRNA population counts. In particular, randomness in the transcription process is captured through a two-state model, where the promoter of a gene stochastically switches between an active and inactive state. High levels of transcription occur from the active state, while the inactive state allows for a low basal rate of transcription. For the two-state model we derive exact analytical formulas for the steady-state mRNA probability distribution and the mRNA auto-correlation function. These results are applied to recent data from the Human Immunodeficiency Virus (HIV) system. Using Akaike Information Criterion (AIC) we select the most likely stochastic model for the transcription process given mRNA histogram data. For the selected model, maximum likelihood estimates of the different kinetic rates associated with the viral promoter are inferred. Analysis reveals that the viral promoter resides mostly in the inactive state and there is a 100-fold difference in the rate of mRNA synthesis from the active and inactive state. In summary, formulas presented here are an important resource for reverse engineering genetic promoters from single-cell mRNA copy number data.