Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^d$ with radius $r$ as $B(x, r) \colon = \{ y\in \mathbb{R}^d, \| y - x\|_2 \leq r \} .$ We want to use the Euclidean balls centered at $X_1, \ldots,X_n \ldots$ to cover $S$. Define the random covering number $N$ as $$ N = \inf \bigl\{ n\colon S \subset \cup_ {i=1}^n B(X_i, r) \big\} , $$ where $r\in (0,1)$.

I was wondering if there is a way to derive the distribution of $N$. If this is too hard, can we at least derive some bound on the tail probability of $N$? And how does $N$ depends on the dimensionality $d$ and the radius $r$?

If we can characterize $N$ for $S = [0,1]^d$, can we do similar things for general sets in Banach spaces?