# 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

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