  A wide range of physical problems can be described by randomly-oriented
linear trajectories, including any system of objects, organisms, particles, or
rays that follow a linear path. Dependent upon the particular random variables
that define translation and direction, a description of the probabilistic
behavior through a static volume can reveal evidence on an object's physical
properties. Yet more information may be available in cases where this behavior
is a function of volume shape. In this study two types of random trajectories
are considered as they pass through a box of arbitrary relative dimension. One
type defines trajectories from uniform random selection of a spatial location
paired with a directional vector; the other is formed from a uniformly
distributed position vector on a surface. The joint probability distributions
for trajectory length as a function of box position are formulated and then
examined for different size boxes, and their physical representations
discussed.
