The Kalman filter is a powerful state estimation tool for linear dynamic systems with Gaussian noise, especially effective in discrete state space. In this paper, we propose a design procedure and performance analysis of an optimal linear Kalman filter applied to a one-dimensional dynamic system with random noise satisfying Markov properties. The system model is built in an extended state space including angle, angular velocity, angular acceleration, and system bias, with measurements observing only a portion of the state. The system is discretized using the Euler method for numerical simulation implementation. The Kalman algorithm is developed closely based on probabilistic assumptions, including white Gaussian noise and time-independent Markov assumption. By repeatedly simulating with randomly generated noise, we evaluate the mean square error (MSE) of the estimated states and examine the convergence behavior of the filter. The results show that the Kalman filter converges stably, significantly reduces estimation error, and meets the optimal estimation requirements in realistic noisy environments.