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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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  • $\begingroup$ Do you mean the circle interior, or the circumference? $\endgroup$
    – leonbloy
    Jun 3, 2016 at 13:16
  • $\begingroup$ I mean the interior $\endgroup$ Jun 3, 2016 at 13:18
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    $\begingroup$ 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. $\endgroup$
    – Ian
    Jun 3, 2016 at 13:50
  • $\begingroup$ 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. $\endgroup$
    – DanielV
    Jul 15, 2020 at 7:47
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    $\begingroup$ Does this answer your question? Average distance between two points in a circular disk $\endgroup$
    – Toby Mak
    Jul 15, 2020 at 7:58

1 Answer 1

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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:

  • $F_{S|X}(s|x)=P(S \ge s | x)$

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

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

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  • $\begingroup$ Thanks, I think this is a good approach $\endgroup$ Jun 3, 2016 at 13:52

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