Suppose $(X,Y)$ is bivariate normal with $\mu=\begin{pmatrix} 0 \\ 0 \end{pmatrix}$, $\Sigma=\begin{pmatrix} \sigma^2 & \rho \sigma^2 \\ \rho \sigma^2 & \sigma^2 \end{pmatrix} $
Is it possible to show that for some $\sigma>0,0<\rho<1$, it holds for every $y\in \mathbb R$,
$$x- E(Y|Y\leq y, X=x)$$
is nondecreasing in $x$ for all $x\in \mathbb R$?
I write the formula as
$$x-\rho x+ \sqrt{1-\rho^2}\sigma \frac{\phi(\frac{y-\rho x}{ \sqrt{1-\rho^2}\sigma})}{\Phi(\frac{y-\rho x}{ \sqrt{1-\rho^2}\sigma})}$$
where $\Phi$ and $\phi$ are cdf and density of the standard normal distribution. Is there any way to further simply the formula?
Thanks.