[en] We study the position of a biased tracer particle (TP) in a bath of hardcore particles moving on a lattice of arbitrary dimension and in contact with a reservoir. Starting from the master equation satisfied by the joint probability of the TP position and the bath configuration and resorting to a mean-field-type approximation, we presented a computation of the fluctuations of the TP position in a previous publication (Bénichou et al 2013 Phys. Rev. E 87 032164). Counter-intuitively, on a one-dimensional lattice, the diffusion coefficient of the TP was shown to be a nonmonotonic function of the density of bath particles, and reaches a maximum for a nonzero value of the density. Here, we (i) give the details of this computation and offer a physical insight into the understanding of the nonmonotonicity of the diffusion coefficient; (ii) extend the mean-field-type approximation to decouple higher-order correlation functions, and obtain the evolution equation satisfied by the cumulant generating function of the position of the TP, valid in any space dimension. In the particular case of a one-dimensional lattice, we solve this equation and obtain the probability distribution of the TP position. We show that the position rescaled by its fluctuations is asymptotically distributed according to a Gaussian distribution in the long-time limit. (paper)