Conditional expectation of multivariate normal distribution with inequality condition

Question

Let $X_1, X_2, X_3$ be jointly normal with means $\mu_1,\mu_2,\mu_3$ and covariances $\sigma_{ij}$. I know that $$ \mathbb{E}[X_1\mid X_2=x_2]=\mu_1+\frac{\sigma_{12}}{\sigma_2^2}(x_2-\mu_2) $$ and $$ \mathbb{E}[X_1\mid X_2<x_2]=\mu_1-\sigma_{12}\frac{\phi(\frac{x_2-\mu_2}{\sigma_2})}{\Phi(\frac{x_2-\mu_2}{\sigma_2})} $$ where $\phi$ and $\Phi$ are the density and probability distribution functions for the standard normal distribution. Is there a similar expression for $$ \mathbb{E}[X_1\mid X_2=x_2,X_3<x_3]~? $$ Thank you in advance.

Answer 1

For $\operatorname E(X_1\mid X_3<x_3)$ I'm getting $$ \operatorname E(X_1\mid X_3<x_3) = \mu_1 - \frac{\sigma_{13}}{\sigma_3} \cdot \frac{\varphi\left( \dfrac{x_3-\mu_3}{\sigma_3} \right)}{\Phi\left( \dfrac{x_3-\mu_3}{\sigma_3} \right)}. \tag 1 $$ Once you're given $X_2=x_2,$ you have a certain conditional probability distribution of $(X_1,X_3),$ which is bivariate normal with expected value given by \begin{align} & \operatorname E(X_1\mid X_2=x_2) = \mu_1 + \frac{\sigma_{12}}{\sigma_2^2} ( x_2-\mu_2 ), \\[10pt] & \operatorname E(X_3\mid X_2=x_2) = \mu_3 + \frac{\sigma_{32}}{\sigma_2^2} ( x_2-\mu_2 ), \end{align} and variance given by \begin{align} & \operatorname{var}(X_1\mid X_2=x_2) = \sigma_1^2 - \frac{\sigma_{12}^2}{\sigma_2^2}, \\[10pt] & \operatorname{var}(X_3\mid X_2=x_2) = \sigma_3^2 - \frac{\sigma_{23}^2}{\sigma_2^2}, \\[10pt] & \operatorname{cov}(X_1,X_3\mid X_2=x_2) = \sigma_{13} - \frac{\sigma_{12} \sigma_{23}}{\sigma_2^2}. \end{align} Now apply line $(1)$ above but with these conditional expectations and conditional variances. Where you had $\sigma_{13}$ in line $(1),$ you'll have $\sqrt{\sigma_{13} - \frac{\sigma_{12} \sigma_{23}}{\sigma_2^2}},$ etc.