Dynamic interactions with caregivers are essential for infants to develop cognitive abilities, including aspects of action, perception, and attention. We hypothesized that these abilities can be acquired through the predictive learning of sensory inputs including their uncertainty (inverse precision) in terms of variance. To examine our hypothesis from the perspective of cognitive developmental robotics, we conducted a neurorobotics experiment involving a ball-playing interaction task between a human experimenter representing a caregiver and a small humanoid robot representing an infant. The robot was equipped with a dynamic generative model called a stochastic continuous-time recurrent neural network (S-CTRNN). The S-CTRNN learned to generate predictions about both the visuo-proprioceptive states of the robot and the uncertainty of these states by minimizing a negative log-likelihood consisting of log-uncertainty and precision-weighted prediction error. The experimental results showed that predictive learning with uncertainty estimation enabled the robot to acquire infant-like cognitive abilities through dynamic interactions with the experimenter. We also discuss the effects of infant-directed modifications observed in caregiver-infant interactions on the development of these abilities.