In this paper, we present theoretical developments for the analysis and design of linear dynamic process systems in the presence of uncertain parameters, described by Gaussian probability distribution functions. The concept of the stochastic flexibility index, as a metric for quantifying the ability of a process to maintain feasible operation in the face of stochastic uncertainties, is first extended to linear dynamic models. A detailed procedure for evaluating the stochastic flexibility over time, is then outlined. This procedure fully exploits the mathematical properties of the system to enable derivation of analytical expressions describing the dynamic feasible region. These expressions are then used as constraints in a single-stage design formulation for the determination of an economically optimal process design that meets a desired stochastic flexibility target over the entire time horizon of interest. With this formulation, the trade-offs between cost and target flexibility can be investigated. A numerical example is used to demonstrate the potential of the techniques.