Random chord distributions and containment functions

Formulas for the probability distribution function of the length of random chord in a convex body intersected by a random line are established by using the containment function and the chord projection function of the convex body. Explicit expressions of random chord distributions for several special convex bodies are given. An isoperimetric-type inequality for the moments of random chord is proved. Similar results for random distance distribution of two random points in a convex body are also obtained.