Stochastic ordering

Summary

A stochastic ordering compares two random variables through their distributions, not through their values. The basic one is the usual stochastic order. In it, $X$ is stochastically smaller than $Y$ when $X$ is less likely than $Y$ to exceed every threshold. Equivalently, there is a coupling of the two laws in which $X \le Y$ holds pointwise. Other orders, such as the convex order and the hazard rate order, compare variability or ageing. They are used in reliability theory, queueing, economics and decision theory.