I am trying to figure out the expected angle that a random point makes with a given line.
More formally, let $l$ be a fixed line and $P$ be a set of $n$ points in d-dimensional Euclidean space. Furthermore, let $o$ be a fixed point on $l$. Assume that no point of $P$ is in $l$. Let $x$ be an uniformly random sample from $P$. For any point $p \in P$, let $p_l$ be its orthogonal projection on $l$.
We have that
$E[dist(x,x_l)] = \frac{1}{|P|}\sum_{p\in P}dist(p,p_l)$.
$E[dist(o,x_l)] = \frac{1}{|P|}\sum_{p\in P}dist(o,p_l)$.
Note that these two random variables are not independent. How can the quatity $E[\frac{dist(x,x_l)}{dist(o,x_l)}]$ be computed?
More generally , given two non-negative random variables $X,Y$ ($Y$ having nonzero expectation), how can we compute $E[\frac{X}{Y}]$?
It would be $$\frac{1}{|P|} \sum_{p \in P} \frac{d(p,p_l)}{d(o,p_l)}.$$
More generally, if $X$ and $Y$ are discrete random variables, then $$E[X/Y] = \sum_{x,y} \frac{X}{Y} P(X=x, Y=y).$$ In your case, $X=d(x,x_l)$ and $Y=d(o, x_l)$ with joint distribution $P(X=d(p,p_l), Y=d(o,p_l)) = 1/|P|$ for $p \in P$.