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  "body": "\n## Summary\n\nA stochastic ordering compares two random variables through their distributions, not through their values.\nThe basic one is the usual stochastic order.\nIn it, $X$ is stochastically smaller than $Y$ when $X$ is less likely than $Y$ to exceed every threshold.\nEquivalently, there is a [[coupling|HTR7J0W66V]] of the two laws in which $X \\le Y$ holds pointwise.\nOther orders, such as the convex order and the hazard rate order, compare variability or ageing.\nThey are used in reliability theory, queueing, economics and decision theory.\n\n## Explanations\n\n### L1 human\n\nOne lottery is stochastically better than another if, for every amount of money, it is at least as likely to pay that amount or more.\nFor example, a waiting time with an [[exponential|0E99F51GVB]] law of rate $2$ is stochastically smaller than one with rate $1$, because $e^{-2t} \\le e^{-t}$ for all $t \\ge 0$.\nTwo bell curves centred at $0$ with different spreads are not comparable in this order.\nEach one is more likely than the other to exceed some thresholds.\n",
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