This paper derives a complete analytical solution for the probability distribution of the configuration of a non-holonomic mobile robot that moves in two spatial dimensions by satisfying the unicycle kinematic constraints. The proposed solution differs from previous solutions since it is obtained by deriving the analytical expression of any-order moment of the probability distribution. To the best of our knowledge, an analytical expression for any-order moment that holds even in the case of arbitrary linear and angular speed, has never been derived before. To compute these moments, a direct integration of the Langevin equation is carried out and each moment is expressed as a multiple integral of the deterministic motion (i.e., the known motion that would result in absence of noise). For the special case when the ratio between the linear and angular speed is constant, the multiple integrals can be easily solved and expressed as the real or the imaginary part of suitable analytic functions. As an application of the derived analytical results, the paper investigates the diffusivity of the considered Brownian motion for constant and for arbitrary time-dependent linear and angular speed.