Statistical hydromechanics originated by Hopf is generalized to take into account the influence of an external random force field with an arbitrary (not necessarily Gaussian) distribution. Starting from the space-time functional formalism for turbulence recently established, a general expression for the characteristic functional describing the statistical behavior of fluid at each instant is derived in the form of a functional integral. For the case that the random force field is temporally correlated in the delta-function manner, a functional-differential equation is obtained which governs the time-evolution of the characteristic functional. This equation, being a counterpart of the Hopf equation which was given for the case of no random excitation, involves Novikov's equation, as a special case.