Given a set $\{X_1,\ldots,X_N\}$ of real i.i.d. random variables, drawn from a common parent pdf $p_X(x)$, what is the probability that, given one random variable taking value in $(t-dt,t)$, there are no other random variables in the interval $(t,t+s)$ and there is another random variable in $(t+s,t+s+ds)$? In other words, what is the probability of an empty gap of size $s$ to the right of the point $t$ on the real axis, as a function of $s,t$ (given $N$ and $p_X(x)$)?
Is there a general formula for finite $N$, and/or an asymptotic formula for large $N$?
The book by Mehta on random matrices reports an asymptotic argument on pag. 11-12 that does not mention/require either the parent pdf or the point $t$, and the gap pdf turns out to be exponential, $\exp(-\rho s)\rho ds$, where $\rho$ is related to the "average density of points", however the argument is very heuristic and not terribly clear in my view. Is there a way to formalize it a bit better? Many thanks, folks.