We study the Rayleigh-Benard stability problem for a fluid confined within a square enclosure subject to random perturbations in the temperature distribution at both the horizontal walls. These temperature perturbations are assumed to be non-uniform Gaussian random processes satisfying a prescribed correlation function. By using an accurate Monte Carlo method we obtain stochastic bifurcation diagrams for the Nusselt number near the classical onset of convective instability. These diagrams show that random perturbations render the bifurcation process to convection imperfect, in agreement with known results. In particular, the pure conduction state does no longer exist, being replaced by a quasi-conduction regime. We have observed subcritical and nearly supercritical quasi-conduction stable states within the range of Rayleigh numbers Ra = 0-4000. This suggests that random perturbations in the temperature distribution at the horizontal walls of the cavity can extend the range of stability of quasi-conduction states beyond the classical bifurcation point Ra-c = 2585.02. Analysis of the stochastic bifurcation diagrams shows the presence of a stochastic drift phenomenon in the heat transfer coefficient, especially in the transcritical region. Such stochastic drift is investigated further by means of a sensitivity analysis based on functional ANOVA decomposition. (C) 2012 Elsevier Ltd. All rights reserved.