Given the diagram below (The first figure): $$f_r(r)= \frac{\lambda \pi r \exp{(-\lambda \pi r^2/2)}}{1-e^{-\lambda \pi R^2/2}}$$
where $$0<r\leq R$$
The pdf
$$f_{\text{Z}}(\text{z})=\frac{2\lambda\sqrt{R^{2}-\text{z}^{2}}}{1-e^{-\lambda\pi{R}^{2}/2}} e^{-\lambda{R}^{2}(cos^{-1}(\frac{\text{z}}{R})-(\frac{\text{z}}{R})\sqrt{1-(\text{z}/R)^{2}})}$$
where $$0<z\leq R$$
Question: How can this PDF be conditioned on the area UQWD given in the second figure. Note that in the second figure, the two outer circles have the same radius.
The problem is I know its possible, but I can't picture how this could be done.
