Joint probability density functions of random trajectories through a box

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.