jats:p A method is derived for computing a number of key statistical properties of the vibrational energy of each component of a built-up system that has random properties. The energy is considered to be a random function of frequency, and the derived statistical properties include: the mean rate at which the energy crosses a specified level, the probability that the energy will exceed a specified level within a given frequency band, the mean trough-to-peak height, the rate of occurrence of peaks, and the mean quefrency (a measure of the rate of fluctuation of the energy). The analysis is based on combining statistical energy analysis with a non-parametric model of uncertainty based on the Gaussian orthogonal ensemble and avoids the use of Monte Carlo simulations or large computational models. By way of example, the method is applied to a number of coupled plate systems. </jats:p>