As a theoretical tool, we use the single-mode Fokker-Planck distribution for an isotropic plate to describe the random vibration of thermally buckled composite plates. The Fokker-Plank distribution becomes singular as uniform plate temperature far exceeds, say, 20 times the critical buckling temperature. Then the asymptotic high-temperature moments depend only on the snap-through displacement, testifying that stochastic dynamics has degenerated into a static snap-through problem in the limit of high plate temperature and large temperature gradient across the plate thickness. Otherwise, it is nonsingular and bimodal for low and moderate plate temperatures. From the nonsingular Fokker-Planck distribution, we have deduced peak scaling by the standard deviation of displacement distribution and derived a functional form for strain distribution by using the quadratic relation between strain and displacement. They have been validated by the displacement histograms of numerical simulations and the strain histograms of thermally buckled plate experiments.