[Math] the average distance between two random points inside a circle

Question

Assume you have a circle with some radius r. What is the average distance between two random points inside the circle?

(Edit: This is different from this already answered question, because here the points are inside the circle area, not on the circle circumference.)

Answer 1

Sketch. Let $S$ be the distance between the points, and let $X$ be the distance of the first point from the center of the circle.

Then compute:

• $F_{S|X}(s|x)=P(S \ge s | x)$

• $F_S(s)=P(S \ge s ) = \int F_{S|X}(s|x) \, f_X(x) \, dx$

• $E(S) = \int_0^\infty (1-F_S(s)) ds$