TITLE: probability of strong correlated random events
QUESTION [2 upvotes]: Let $A=\{X\leq \text{mean}(X)\}$, $B=\{Y\leq \text{mean}(Y)\}$, correlation coefficient $\rho(X,Y)=-0.6$,
$\text{Prob}(A)=1/2+\epsilon$, $\text{Prob}(B)=1/2-\epsilon$ for some small $\epsilon$.
any good bound for
$$\text{Prob}(AB)\leq ?$$
none of pdf of $X$ and $Y$ known.
thanks

REPLY [1 votes]: There can't be any bounds better than $0 < P(AB) < P(B)$.  Consider cases where with very high probability $X$ and $Y$ are close to $0$: in this region
you could have at one extreme $\{Y \le 0\} \subseteq \{X \le 0\}$ or at the other extreme $\{X \le 0\} \cap \{Y \le 0\} = \emptyset$.  With very small probability $X$ and $Y$ have very large values, and these can be 
adjusted so that the means are $0$ and the correlation is $-0.6$.
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