# Points uniformly distributed on a circle

We select randomly two points in the circumference (with length equal to 1) of a circle. Let $$X,Y$$ be those points (independent and uniformly distributed) and $$D$$ the arc distance between them. Since we move anti clock-wise, we select first $$X$$ then $$Y$$. Looking for $$E[D]$$.

My intuition says it might be $$\frac{\pi}{4}$$. Am I correct?

• The points determine $2$ arcs. Is $D$ the length of the shortest arc? – drhab Apr 10 at 13:18
• @drhab it is written as arc distance so I assume it is the smallest by definition – Μάρκος Καραμέρης Apr 10 at 13:21
• D is the length of the arc from X to Y – EmKal Apr 10 at 22:28
• dropbox.com/s/7n4c4e1e5k3qmkz/circle9.JPG?dl=0 – EmKal Apr 10 at 22:51
• The correct answer is $\frac{1}{2}$. We can figure this also by using symmetry. – EmKal Apr 12 at 17:50