We investigate the time evolution of the mean location and variance of a charged particle subject to random collisions that are Poisson distributed. The particle moves on a plane and is subject to a magnetic field applied perpendicular to the plane, so it is constrained to move in circles in the absence of collisions. We develop a procedure that yields analytic expressions of the mean and variance. These results are valid for arbitrary times after the start of the walk, including early on when, on average, less than one collision is expected. As an example of their applicability, we use these expressions to model experimental results and simulations of suprathermal ions propagating in a turbulent plasma in TORPEX (the TORoidal Plasma EXperiment).