Probability density function for radius within part of a sphere I would like to find the probability density function for radius within a given section of a sphere.  For example, suppose I specify  $\pi / 4 < \theta < \pi / 3$ and $\pi /7 < \phi < \pi /5 $ and $1 < r < 4$.  If I select a point at random from within this region, what is the probability distribution of the resulting values?

Of course the probability that the radius of a randomly chosen point has a certain given value is zero. But we can sensibly talk about the probability density function $r\mapsto f_R(r)$ which is relevant in this example.
We are given a certain domain $A\subset S^2$ in terms of geographical angles, and the random point ${\bf x}$ in question has to satisfy ${\bf x}=r{\bf u}$ with  ${\bf u}\in A$ and $1\leq r\leq 4$. Let $|A|$ be the spherical area of the domain $A$.
Using Fubini's theorem we see that the volume of the set $S_r\subset{\mathbb R}^3$ of admissible points ${\bf x}$ with $|{\bf x}|\leq r$ is given by
$${\rm vol}(S_r)=|A|\int_1^r r^2\ dr={r^3-1\over 3}\>|A|\ ,$$
and the probability that $|{\bf x}|\leq r$ is given by
$$F(r):={{\rm vol}(S_r)\over{\rm vol}(S_4)}=
{r^3-1\over 63}\qquad(1\leq r\leq4)\ .$$
It follows that
$$f_R(r)={d\over dr}F(r)={r^2\over21}\qquad(1\leq r\leq 4)\ .$$
