Random copy of a segment within a convex domain

Abstract

Under the assumption that S is a segment of the length l and D is a bounded, convex domain in the Euclidean plane ℝ2, the paper considers the randomly moving copy L of S, under the condition that it hits D. Denote by L| the length of L ∩ D. In the paper an elementary expression for the distribution function F L (x) of the random variable L| is obtained. Note that F L (x) can have a jump at the point l or can be a continuous function depending on l and the domainD. In particular, a relation between chord length distribution functions of D and F L (x) is given. Moreover, we derive explicit forms of F L (x) for the disk and regular n-gons with n = 3÷7.