A bridge, idealized as an Euler beam, is assumed to have an irregular road profile. The model of a vehicle is a single-degree-of –freedom oscillator moving with constant horizontal velocity. The road profile is modelled as a Gaussian stochastic process governed by a set of two stochastic differential equations driven by a Wiener process. A single-term normal mode expansion is assumed for the beam response. The equations of motion are derived from Lagrange’s equations. The augmented state vector of the problem is governed by a set of first order stochastic differential equations with time dependent coefficients. The differential equations for mean values and cross-covariances of the state variables are obtained and solved numerically. The effect of different parameters , such as e.g. the vehicle-to-bridge mass ratio, the level of the stochastic road profile irregularity and the horizontal velocity of the vehicle, on the response of the bridge and of the vehicle is analysed.