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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.)

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Kent Munthe Caspersen
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  • Do you mean the circle interior, or the circumference? – leonbloy Jun 03 '16 at 13:16
  • I mean the interior – Kent Munthe Caspersen Jun 03 '16 at 13:18
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    I don't think this is a duplicate of that question because that appears to be about points on the circle whereas this is about points in the circle. – Ian Jun 03 '16 at 13:50
  • For this problem it really matters how you are choosing the points. If you are choosing them with uniform distribution relative to a cartesian plane, or if you are choosing them with uniform radius and uniform angle. Those are not the same thing. – DanielV Jul 15 '20 at 07:47
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    Does this answer your question? Average distance between two points in a circular disk – Toby Mak Jul 15 '20 at 07:58
  • @TobyMak It is definitely the same question, but the solution is a link to a broken link to a missing external pdf, so not really a solution. I think that kind link-only answer is discouraged, not sure how it affects dups though. – DanielV Jul 15 '20 at 08:02
  • Well, there is another solution below the one you mentioned where the proof does not rely on any external links. I don't think the answers are lacking in any way, such that they are incomplete without using external links, so closing this question as a duplicate is fine. – Toby Mak Jul 15 '20 at 08:04

1 Answers1

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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:

  • $F_{S|X}(s|x)=P(S \ge s | x)$

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

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

leonbloy
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