Gene transcription and protein synthesis can be widely found in biological research. Based on this viewpoint, this article focuses on a two-dimensional stochastic delay differential equation (SDDE) to investigate the gene expression. A wealth of dynamic behaviors from a mathematical perspective are observed. An optimized method to handle the stochastic delayed system is proposed and we applied it to the model in this paper. Then an averaging Ito equation is obtained. Based on this, we studied the stochastic bifurcation of the system and the corresponding probability distribution with time delay, degradation rate, the intensity of the stochastic process, and the randomization degree as parameters. The numerical results revealed the transition of the probability density function of the system from a Dirac δ function to a nontrivial stationary state. In addition, stochastic P-bifurcation occurs when the presentation of the stationary probability density change from singular peak to crater.