Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 30 / 3 / a / Solution

Order configurations coordinatewise: $\omega \le \eta$ means $\omega (e)\le \eta (e)$ for every edge. Then stochastic domination is the conditionEquivalently, ${\mathbb{E}}_{{\mu }_1}f\ge {\mathbb{E}}_{{\mu }_2}f$ for every real-valued order-preserving function $f$. The equivalence follows on this finite space by writing $f$, up to an additive constant, as a nonnegative linear combination of indicators of its increasing upper level sets. A coupling of probability distributions supported on pairs $(X_1,X_2)$ with $X_1\ge X_2$ proves this domination; conversely such an ordered coupling of probability distributions exists on a finite partially ordered space.