Suppose $X$ and $Y$ are jointly normal with mean $(\mu_X,\mu_Y)^T$ and covariance matrix $\begin{pmatrix} \sigma_x^2 & \rho \\ \rho & \sigma_Y^2\end{pmatrix}$. How can I compute the expectation and variance of $X|Y>0$?
Is it valid to write $\mathbb{E}(X|Y>0) = \int_0^\infty \mathbb{E}(X|Y=y)f_Y(y) dy$, where $f_Y$ is the marginal density of $Y$?
