Lateral vibrations are widely recognized as the leading cause of drill-string failures in oil well operations. The finite elements method formulation is used for the mathematical modelling. Due to the erratic nature of the forces at the drill-bit, a stochastic dynamics approach is followed. Both the method of statistical linearization, and the Monte Carlo method are used to analyze the system dynamics. Numerical results pertaining to data obtained by measurement while drilling tools are presented. The significance of the present study, over studies available in the literature, hinges upon the computational efficiency of the adopted piecewise linear model of the drill-string---well formation interaction and the derivation of information on the probability density function, and the power spectrum of the erratic drill-string motion. It is expected that this study will enhance the interest in using stochastic dynamics techniques in drilling system analysis and design.