# Can we say “X smaller than Y” in probability if $P\{X < a\} \ge P\{Y < a\}$?

I have an impression that there is specific way to describe the relationship between 2 such variables, but I can remember exactly. And I also remember there should be an special notation for this, but I can't find it by Google either.

Is there really such a term and a notation used by mathematicians or it's just my illusion?

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Should't the two constants $a,b$ in your title be equal? – Alex Becker Mar 20 '12 at 23:37
Yes, they should be equal. – ablmf Mar 20 '12 at 23:39

## 1 Answer

If it's true of all values of $a$, then one says $X$ is stochastically smaller than $Y$.

One place where the concept shows up is in statistical hypothesis testing. One normally speaks of comparing two probability distributions rather than of comparing two random variables. Say one is testing the null hypothesis that a die is fair, using the usual chi-square test. If the die is not fair, then the test statistic has non-central chi-square distribution rather than a (central) chi-square distribution. The latter is stochastically smaller than the former. Hence it makes sense to reject the null hypothesis if the test statistic is large.

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I would like to add that $X \le Y$ in a stochastic ordering has nothing to do with $X(\omega)\le Y(\omega)$. In most notations, a s.o. is written over the $\le$ sign to make this distinction (which is crucial from a beginner's perspective). – Bravo Mar 21 '12 at 6:11