Conside two non-independent gaussian random variables: $$(,)∼\text{BiNormal}[(,),(,),]$$
I'm interested in understanding when (i.e. for which values of the distribution parameters) it's true that $$\frac{\partial E[T |C<f(T)+\tau]}{\partial \tau}>0$$ where $f(T)>0$ and $f'(T)>0$ for all $T\in \Bbb R$.
Is there a way to parametrise this conditional expectation, in order to sign the derivative?
NOTES:
In case it's helpful: I'm actually interested specifically in $f(T)=b*\Phi\left(\frac{T-\mu_a}{\sigma_a}\right)$, where $\Phi(\cdot)$ is the CDF of the Standard Normal distribution, and $b\in \mathbb{R}^{+}$ is a constant.
Here is a relevant approach for a simpler version of this question: Parameterising conditional expectation of gaussian random variables*
I've tried spelling out the expectation via integrals over the joint distribution, and then taking derivatives in Mathematica. This approach was unsuccessful and I posted about it here: https://mathematica.stackexchange.com/questions/222708/derivative-for-conditional-expectation-of-non-independent-gaussian-variables-is