# Conditional expectation for multivariate normal distribution

Assume $$\mathbf{X}=[X_1,X_2,X_3]$$ follows a multivariate normal distribution, $$\mathbf{X}\sim N_3 (\mu,\Sigma)$$, where $$\text{cov}(X_i,X_j)\neq 0$$, for $$i\neq j$$. What is $$\mathbb{E}[X_1 X_2|a_1X_1+a_2X_2+a_3 X_3>t]$$ for any arbitrary number $$a_i,t\in \mathbb{R}$$?

• I don't think this will have an exact form – Stan Tendijck Apr 14 at 19:21
• I could calculate the exact solution for bivariate case, $E[X_1 X_2 | X_1+X_2 >t]$. Any good approximation will also work for me! – RaTa Apr 14 at 23:15