Sensory-motor systems can extract statistical regularities in dynamic uncertain environments, enabling quicker responses and anticipatory behavior for expected events. Anticipatory smooth pursuit eye movements (aSP) have been observed in primates when the temporal and kinematic properties of a forthcoming visual moving target are fully or partially predictable. To investigate the nature of the internal model of target kinematics underlying aSP, we tested the effect of varying the target kinematics and its predictability. Participants tracked a small visual target in a constant direction with either constant, accelerating, or decelerating speed. Across experimental blocks, we manipulated the probability of each kinematic condition varying either speed or acceleration across trials; with either one kinematic condition (providing certainty) or with a mixture of conditions with a fixed probability within a block. We show that aSP is robustly modulated by target kinematics. With constant-velocity targets, aSP velocity scales linearly with target velocity in blocked sessions, and matches the probability-weighted average in the mixture sessions. Predictable target acceleration does also have an influence on aSP, suggesting that the internal model of motion that drives anticipation contains some information about the changing target kinematics, beyond the initial target speed. However, there is a large variability across participants in the precision and consistency with which this information is taken into account to control anticipatory behavior.