Minimum Distance Distribution of two Uniformly Distributed Samples I am working with genome data, more precisely two kinds of DNA motifs distributed over a large DNA sequence. I am searching for the probability distribution of the minimum distances between the two motifs, assuming that they are distributed randomly and independently of each other.
The problem is analogous to the minimum distance distribution of two uniformly distributed random samples (with different sizes) distributed over a fixed interval of length $L$.
It would be easy to run simulations of the problem but I would guess that this problem is simple enough for an exact solution.  (Accordingly, I feel a little dumb that I can not come up with a good solution).
On Google I found exact solutions for two randomly distributed points, which would be the extreme case where both samples have size 1, but I need a more general solution.
Something like "the problem is not exactly solvable" would also help, if reliable.
 A: If what you ask is to find the distribution of $Z=\min|x_i-y_j|$ where $x_i, i=1,\dots,m$ and $y_j, j=1,\dots,n$ are independent random points on $[0,1]$ (scaling to $L$ is trivial), then one can write an exact formula but it is a bit ugly:
$$
P(Z>t)=\frac{m!n!}{(m+n)!}\sum_{k\ge 1}A(m,n,k)(1-kt)_+^{m+n}
$$
where $A(m,n,k)=2{m-1\choose \ell-1}{n-1\choose\ell-1}$ if $k=2\ell-1$ and 
$A(m,n,k)={m\choose \ell}{n-1\choose\ell-1}+{m-1\choose \ell-1}{n\choose\ell}$ if $k=2\ell$.
The explanation is very simple: you just condition upon $k$ transitions between $x$'s and $y$'s, count the numbers of the corresponding arrangements and note that, by removing an interval of length $t$ in every transition, you effectively restrict the sidelength of the available cube (or simplex, if you prefer to think of the ordering as fixed at this moment) to $(1-kt)_+$.
This is as good as it gets if $min(m,n)$ is small. If that minimum is large, then you are better off with asymptotic analysis. Here is the back of envelope computation for $m,n$ large and comparable (I prefer to avoid assigning the exact meaning to the last word now because I'm not even sure this requirement is really needed).
You can think just of two independent Poisson processes on $[0,1]$ with intensities $m,n$. The union is still a Poisson process, so the intervals are independent exponential random variables $I_j$ with the law $P(I_j>t)=e^{-(m+n)t}$. Now the typical number of "interesting intervals" (transitions between $x$ and $y$ is just $(m+n)P$ where $P=\frac{2mn}{(m+n)^2}$ is the probability that two endpoints are from different processes (the splitting of the union into two parts is independent of the sampling). Taking the minimum of a bunch of independent exponential distributions is a piece of cake, so if you put everything together, you get 
$$
P(Z>t)\approx e^{-2mnt}\,,
$$
which (I hope) should agree decently with numerics.
