Paper 2, Section II, D

Let $\Delta$ be the disc of radius 1 with centre at the origin $O$. Let $P$ be a random point uniformly distributed in $\Delta$. Let $(R, \Theta)$ be the polar coordinates of $P$. Show that $R$ and $\Theta$ are independent and find their probability density functions $f_{R}$ and $f_{\Theta}$.

Let $A, B$ and $C$ be three random points selected independently and uniformly in $\Delta$. Find the expected area of triangle $O A B$ and hence find the probability that $C$ lies in the interior of triangle $O A B$.

Find the probability that $O, A, B$ and $C$ are the vertices of a convex quadrilateral.

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