[math]E(X|a \leq Y \leq b)[/math]

where [math]c \leq X \leq d [/math], [math]X[/math] and [math]Y[/math] are (doubly truncated) Gaussians with the same mean and different variance, and [math]a < c < d < b[/math] are the truncation points.

To start off, I wrote:

[math]E(X|a \leq Y \leq b) = \int_c^d x f_{X|c \leq Y \leq d}(x)dx [/math]

where

[math]f_{X|c \leq Y \leq d}(x) = \frac{f_{X}(x)}{Pr(c \leq Y \leq d)} =

\frac{\frac{\frac{1}{\sigma_x}\phi (\frac{x-\mu_x}{\sigma_x})}{\Phi(\frac{d-\mu_x}{\sigma_x}) - \Phi(\frac{c-\mu_x}{\sigma_x})}}{\Phi(\frac{d-\mu_y}{\sigma_y}) - \Phi(\frac{c-\mu_y}{\sigma_y})}, \forall c \leq Y \leq d

[/math]

At this point I'm absolutely stuck. Is what I wrote correct? Is there any other way to derive the result more directly?

I've attached a Pdf as well for you to see the formulas more clearly, if needed.

Any help and advice would be GREATLY appreciated.