This paper presents a novel approach to robustness analysis based on quadratic performance metrics of uncertain time-varying systems. The considered time-varying systems are assumed to be linear and defined over a finite time horizon. The uncertainties are described in the form of real-valued random variables with a known probability distribution. The quadratic performance problem for this class of systems can be posed as a parametric Riccati differential equation (RDE). A new approach based on polynomial chaos expansion is proposed that can approximately solve the resulting parametric RDE and, thus, provide an approximation of the quadratic performance. Moreover, it is shown that for a zeroth order expansion this approximation is in fact a lower bound to the actual quadratic performance. The effectiveness of the approach is demonstrated on the example of a worst-case performance analysis of a space launcher during its atmospheric ascent.