Distribution of Intersection Given Uniform Angle 
We have a ray emanating from the point $(-1,0)$ with the angle $\theta$ randomly chosen, uniform on $[-\frac{\pi}{2},\frac{\pi}{2}]$, so the terminal point of the ray hits somewhere on the $y$ axis. What is the distribution of the point of intersection of the ray with the y axis? 

Note that $Y=\tan \theta,$ so that the cumulative distribution for $Y$ is
$$F(y)=P(Y \le y)=P(\tan \theta \le y)=P(\theta \le \tan^{-1}y).$$
Since $\theta$ is uniform on $(-\pi/2,\pi/2)$ the last probability here is
$$F(y)=\frac{\tan^{-1}y+\pi/2}{\pi},$$
which is the cumulative distribution for $Y$ on $\mathbb{R}$. For the density $f(y)$ if needed, take the derivative to get
$$f(y)=\frac{1}{\pi}\frac{1}{1+y^2}.$$
