Probability density function for the radius half the length of two equal intersecting circles Considering this diagram, assuming a uniform distribution in the area of  UQWD, it is still not clear how the probability density function of r becomes $l(r)/S$. Where S is the area of UQWD. What is the proof for the pdf in this case? A schematic diagram is attached below for clarity.
NB: I have taken time to go through this but I haven't gotten a strong clue yet. 

You can parametrize the area UQWD in polar coordinate as
$$
{\cal A} = \{(r, \theta)| r\in [0, r_m], \theta\in [\pi-l(r)/(2 r), \pi+l(r)/(2 r)]\}
$$
For a given $r_0\in [0, r_m]$, one has
$$
P(r\le r_0) = \frac{1}{S}\int_0^{r_0} dr\int_{\pi-l(r)/(2r)}^{\pi+l(r)/(2r)} r d\theta = \frac{1}{S}\int_0^{r_0} l(r) d r
$$
Hence the density $\frac{l(r)}{S}\chi_{[0, r_m]}$.
