By virtue of the mean-square calculus, a technique has been developed to determine the covariance and spectral-density functions for a plate of arbitrary boundary conditions subjected to loadings that are random in space and time. Also cross covariance and resulting spectral density functions are found. As an illustrative example, the theory developed here is applied to a simply supported plate subjected to both a point load and a line load that is random in time only. The solutions are presented in the form of infinite series that converge rapidly.