Found this problem in Grimmett's probability book. I do not know how to properly start it, I tried a few approaches, however they yielded completely different results from what the author's result is.
William Tell is a very bad shot. In practice, he places a small green apple on top of a straight wall which stretches to infinity in both directions. He then takes up position at a distance of one perch from the apple, so that his line of sight to the target if perpendicular to the wall. He now selects an angle uniformly at random from his entire field of view and shoots his arrow in this direction. Assuming that his arrow hits the wall somewhere, what is the distribution function (cdf) of the horizontal distance (measured in perches) between the apple and the point which the arrow strikes.
The main problem that I had with my approaches was that I did not seem to express correctly in terms of the angle, the distance between the apple and the place where the arrow lands.
It is given that the angle (say $\theta$) between the line of shot and the line joining William to the apple is such that $\theta\sim \text{unif}[-\pi/2,+\pi/2]$. The signed horizontal distance from the apple to the point at which the arrow strikes is just $d=\text{tan}(\theta)$, and the magnitude of the distance is just $|d|$. So, the $CDF$ for $|d|$ is just $F(x)=P(|d|\leq x)=P(x\geq \text{tan}\theta \geq -x)=(1/\pi)\int _{-\text{tan}^{-1}(x)}^{\text{tan}^{-1}(x)}d\theta=(2/\pi)\text{tan}^{-1}(x)$.