|Abstract||This research project focuses on random paths with bounded curvature, using differential geometry and statistical mechanics. It is known from the work of Einstein that random walks are connected to Brownian motion and diffusion. We will examine random curves that are not merely continuous but that are smooth and have prescribed bounds on curvature. We examine the distribution of a finite number of endpoints of such random curves. Using Python, we obtain 2-D histograms, graphs, and charts to examine the diffusion resulting from particles moving along these random curves. A central goal in statistical mechanics is to describe the large-scale behavior of systems with the distribution of randomly generated data; we compare the distributions of curve endpoints to the Gaussian (normal) distribution.