Intersections of random line segments

We consider a random collection of n line segments, in which the segments are independently drawn by picking the midpoint according to a given distribution in the plane, by picking a uniformly distributed random direction, and by adding an independently sampled segment length. We study the expected behavior of the number of intersections as a function of n, the lengths of the line segments, and the distribution of the midpoints. This includes a universally applicable law of large numbers.