In coastal areas, variable bottom effects significantly enhance wave nonlinearity and complicate wave propagation. It is of practical interest to characterize the nonlinear effect on the statistics of free surface displacements and particle kinematics. In this work, we take advantage of a fully nonlinear potential flow model to investigate the statistics of unidirectional irregular waves propagating over an uneven bottom. By confronting the simulated results with existing experimental results (free surface elevation and horizontal velocity beneath the mean sea level) in the temporal, spectral, and statistical domains, we show the high fidelity of the model in predicting the nonlinear irregular wave kinematics. As the relative importance of low-frequency harmonics becomes lower for acceleration, the model performance in predicting the measured horizontal acceleration is even better than that for the measured horizontal velocity. The empirical statistical distributions of velocity and acceleration in both horizontal and vertical directions are compared with both the normal (Gaussian) and the log-normal (LN) distributions. The latter requires skewness as an input in addition to the mean and standard deviation of the signal. We notice that, unlike the free surface displacement generally of positive skewness, the signal of velocities and accelerations are sometimes characterized by negative skewness. In such cases, the negative LN distribution should be adopted. Although the LN distribution has rarely been used for short-term statistics of wave elevation and kinematics, the detailed comparisons presented here demonstrate very good performance for all kinematic variables. In particular, in the area following a rapid reduction of water depth, where the sea-state is out-of-equilibrium, the heavy tails in the distributions are well reproduced by the LN model, indicating some generality and merits of this model.