  We consider steady-state current activity statistics for the one-dimensional
totally asymmetric simple exclusion process (TASEP). With the help of the known
operator algebra (for general open boundary conditions), as well as general
probabilistic concepts (for the periodic case), we derive and evaluate
closed-form expressions for the lowest three moments of the probability
distribution function. These are confirmed, to excellent degree of accuracy, by
numerical simulations. Further exact expressions and asymptotic approximations
are provided for probability distributions and generating functions.
