Suppose $(Z\_1, Z\_2)$ is the zero-mean bivariate normal distribution with covariance $\left( \begin{matrix} 1 & \rho; \\ \rho & 1\end{matrix} \right)$ with positive $\rho > 0$. What I want to know a valuable tighy lower bound of the probability
$$\mathrm{P} \left( Z\_1 > z\_1, Z\_2 < z\_2\right)$$
where $z\_1, z\_2$ are some positive constants.


PS:
There are various valuable lower bound for the tail probability $\mathrm{P} \left( Z\_1 > z\_1, Z\_2 > z\_2\right)$ with positive $\rho > 0$, see p495-p499 in [Continuous Bivariate Distributions](https://link.springer.com/book/10.1007/b101765). Can we transform $\mathrm{P} \left( Z\_1 > z\_1, - Z\_2 > -z\_2\right)$ to $\mathrm{P} \left( Z\_1 > z\_1, Z\_2 > z\_2\right)$ keeping the correlation coefficient positive?



The lower bound of bivariate normal distribution
Let $h:=z\_1$, $k:=z\_2$, and $r:=\rho\ge0$. We want to lower-bound $P(Z\_1>h,Z\_2<k)$. Formula (2.11) in the [paper by Willink](https://www.tandfonline.com/doi/abs/10.1081/STA-200031505) (cited the book you linked) gives the following upper bound on $P(Z\_1>h,Z\_2>k)$:  

$$P(Z\_1>h,Z\_2>k) \\ 
\le\Phi(-h)\Big[\Phi\Big(\frac{rh-k}{\sqrt{1-r^2}}\Big)
+re^{(h^2-k^2)/2}\,\Phi\Big(\frac{rk-h}{\sqrt{1-r^2}}\Big)\Big] \tag{1}\label{1}$$
for $h>0$ and states certain optimality properties of this bound; here, as usual, $\Phi$ denotes the standard normal cdf.


Since $P(Z\_1>h,Z\_2<k)=1-\Phi(h)-P(Z\_1>h,Z\_2>k)=\Phi(-h)-P(Z\_1>h,Z\_2>k)$, immediately from \eqref{1} we get the following lower bound on $P(Z\_1>h,Z\_2<k)$:  

$$P(Z\_1>h,Z\_2<k) \\ 
\ge\Phi(-h)\Big[\Phi\Big(\frac{k-rh}{\sqrt{1-r^2}}\Big)
-re^{(h^2-k^2)/2}\,\Phi\Big(\frac{rk-h}{\sqrt{1-r^2}}\Big)\Big]. \tag{2}\label{2}$$
Inequality \eqref{2} will turn into the equality when $r=0$.


