  The curse of dimensionality is a common phenomenon which affects analysis of
datasets characterized by large numbers of variables associated with each
point. Problematic scenarios of this type frequently arise in classification
algorithms which are heavily dependent upon distances between points, such as
nearest-neighbor and $k$-means clustering. Given that contributing variables
follow Gaussian distributions, this research derives the probability
distribution that describes the distances between randomly generated points in
n-space. The theoretical results are extended to examine additional properties
of the distribution as the dimension becomes arbitrarily large. With this
distribution of distances between randomly generated points in arbitrarily
large dimensions, one can then determine the significance of distance
measurements between any collection of individual points.
