Let $X$ and $Y$ be two real-valued random variables, and let $k$ be a real number. Then $$Pr[X>k] \geq Pr[Y>k] - Pr[X \neq Y].$$ Does this inequality have a name, or is there a common reference for it?
Does this probability inequality have a name?
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probability-theory
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0It is true. I doubt it has a name. Note that it is useless when $P[X \neq Y]=1$, which occurs when the joint CDF for $(X,Y)$ is continuous. It is only potentially useful if $P[X\neq Y]<1$. – 2017-01-15
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0Somehow its structure reminds me of "Fano's inequality" from information theory. https://en.wikipedia.org/wiki/Fano's_inequality – 2017-01-15
1 Answers
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It’s probably too straightforward to merit a name: if $Y(\omega) > k$, then either $X(\omega) \neq Y(\omega)$ or $X(\omega) = Y(\omega) > k$. Hence, $[Y > k] \subset [X \neq Y] \cup [X > k]$, so $P(Y>k) \leq P(X>k)+P(X \neq Y)$. I wonder how much this inequality can be of use, since it is basically decomposing $A = (A\cap B) \cup (A \cap B^c)$.