Fix $\epsilon, 0\leq \epsilon\leq 1/2.$ Let $Z\_1,Z\_2$ be zero mean, unit variance Gaussian random variables which are jointly Gaussian with $\mathbb{E}Z\_1Z\_2=-(1-2\epsilon)\leq 0.$


Then, 


$$P(Z\_1Z\_2>0)\geq \epsilon.$$


This curious fact popped out of some calculations I was doing using the Central Limit Theorem. I wonder if it can be proved in some easy way by directly working with the double integral.


I also believe that the inequality is strict when $0<\epsilon<1/2.$ Can this be shown?



On two-dimensional Gaussian integrals
Since $P(Z\_1Z\_2>0)=\frac12-\frac1{\pi}\arcsin(1-2\epsilon)$, indeed $P(Z\_1Z\_2>0)\geqslant\epsilon$, and the inequality is strict except when $\epsilon=0$, $\epsilon=\frac12$ and $\epsilon=1$.


