Stochastic Dominance

Stochastic dominance is an ordering of statistical distributions in which one random variable is at least as likely as another to exceed every threshold. In the usual stochastic order, a real-valued random variable stochastically dominates iff

for every real number . Equivalently, their distribution functions satisfy

The inequality concerns statistical distributions, so it does not require for each outcome in a particular joint realization (Shaked and Shanthikumar 2007).

Stochastic dominance implies for every increasing function for which the two expectation values exist. In particular, it implies the same ordering of their expectation values when these exist, but that ordering alone is insufficient. For example, a random variable taking 0 and 2 with equal probability and a random variable identically equal to 1 have the same expectation value, but neither stochastically dominates the other.