expected distance between two points on the circle

Question

We choose two arbitrary points on the interval $[0,2\pi]$ independently. Treated those points as from a circle with radius equal to $1$, find the expected distance between them (along the chord).

I completely don't know how to start it.

This has already been answered here. – Cecilia Jan 29 '18 at 19:23 — Cecilia, Jan 29 '18 at 19:23

score 1 · Answer 1 · answered Jan 29 '18 at 19:35

1

Let

$$A=(\cos (a),\sin (a)) $$and $$B =(\cos (b),\sin (b)), $$ then

$$d (A,B)^2=1+1-2 \cos (a-b) $$ $$=2 (1-\cos (a-b))=4\sin^2 (\frac {a-b}{2}). $$

thus, the distance between $A $ and $B $ is $$2|\sin (\frac {a-b}{2})|$$

answered Jan 29 '18 at 19:35

hamam_Abdallah

62,951

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I would assume the expected answer should be $4/\pi $. – user Jan 29 '18 at 19:54
@user355705 It must depend on the points. – hamam_Abdallah Jan 29 '18 at 19:59
The expectation value in the sense of probability theory? – user Jan 29 '18 at 20:05

expected distance between two points on the circle

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