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Consider a unit square $OABC$ where,$O\equiv (0,0),A \equiv (1,0),B \equiv (1,1), C \equiv ()$.If two point $P_1$ and $ P_2$is chosen randomly and uniformly in or on the square.how do we find the expected value of angle $\angle P_1OP_2$? My Attempt:- If $(X_1,Y_1)$ and $(X_2,Y_2)$ are the coordinates of the point chosen , Then ezch of the x and y coordiantes follows a uniform distribution in $(0,1)$ .If $\theta $ be the angle $\angle P_1OP_2$ ,then
$$ cos( \theta) =\frac{x_1x_2+y_1y_2}{\sqrt{{(x_1^2+y_1^2)}{(x_2^2+y_2^2)}}}$$. The goal ,then , is to find the pdf of $\theta$.Thanks for any responses/hints in advance

AgnostMystic
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  • Similar problem https://math.stackexchange.com/questions/3629474/average-angle-between-two-randomly-chosen-vectors-in-a-unit-square – Math Lover Dec 01 '20 at 08:31
  • Instead of looking at pdf of angle between them find pdf of angle $\theta$ (with x axis) for any random point first. You will have to consider at $0 \leq \theta \leq \pi/4$ and $\pi/4 \leq \theta \leq \pi/2$ separately. – Math Lover Dec 01 '20 at 08:36

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