In this paper, we apply the previously developed Method of Memory Diagrams (MMD) to the description of an axisymmetric mechanical contact with friction subject to random vibrations. The MMD belongs to a family of semi-analytical methods of contact mechanics originating from the classical Cattaneo-Mindlin solution; it allows one to efficiently compute mechanical and energetic responses to complex excitation signals such as random or acoustic ones. For an axisymmetric contact driven by random normal and tangential displacements having fractal statistical properties, we calculate the friction-induced mechanical energy loss averaged over a large number of realizations. In the considered problem, this energy depends on a very restrained number of parameters: on the rms of random displacements, on the fractal dimension, and on the upper cut-off frequency of the fractal spectrum. In addition, a radial distribution of the dissipated energy has been obtained that has a direct relation to wear in the contact system. For small displacement amplitudes, wear should be expected in an annulus inside of a mean contact circle whereas for large displacements it will start at the contact center.