The theory of residence-time distribution, RTD theory in short, is a cornerstone of chemical engineering science and practice, in general, and that of chemical reactor analysis and design, in particular. The creation of the modern, systematic RTD theory has been attributed to Danckwerts. As evident from his liberal adoption of terminologies of probability and statistics, he was apparently well aware of the stochastic nature of the process that gives rise to a residence-time distribution. While Danckwerts steered the development of the RTD theory essentially along the path of deterministic physics, obviously, the description of RTD is better couched in the statistical or stochastic parlance. Stochastic modeling visualizes the fluid in a flow system as being composed of discrete entities. This visualization reveals a greater insight into the underlying mechanism than deterministic modeling, thereby facilitating our understanding of the flow and mixing characteristic of the system. In the present work, an attempt has been made to derive a unified mathematical model of the RTD during process start-up by rigorously resorting to the theories and methodologies of stochastic processes. Specifically, the expressions for RTDs of molecules, fluid particles or any flowing entities passing through continuous flow systems have been derived from the stochastic population balance of these molecules, particles or entities. The resultant expressions are applicable to both unsteady-state and steady-state flow conditions.