Comparing Chains of Order Statistics

Fix $0\leq k\leq m\leq n$, and let $X_1,...,X_m,Y_1,...,Y_n$ be continuous, independent, and identically distributed random variables. We derive a probability distribution that compares the performance of a $k$-out-of-$m$ system to a $k$-out-of-$n$ system. By virtue of uniformity, we may recast our method of comparison to enumerating lattice paths of a certain exceedance, invoking the Chung-Feller Theorem and Ballot Numbers in our derivation. Another bijection shows that our probability distribution describes the proportion of the first $2k$ steps lying above $x=0$, for a $(m+n)$-step integer random walk, starting at $x=0$ and terminating at $x=m-n$.