I am trying to derive:
$E(X|a \leq Y \leq b)$
where $c \leq X \leq d $, $X$ and $Y$ are (doubly truncated) Gaussians with the same mean and different variance, and $a < c < d < b$ are the truncation points.
To start off, I wrote:
$E(X|a \leq Y \leq b) = \int_c^d x f_{X|c \leq Y \leq d}(x)dx $
where
$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 $
At this point I'm absolutely stuck. Is what I wrote correct? Is there any other way to derive the result more directly?
