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 $a_i,t\in \mathbb{R}$?

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    $\begingroup$ I don't think this will have an exact form $\endgroup$ – Stan Tendijck Apr 14 at 19:21
  • $\begingroup$ 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! $\endgroup$ – RaTa Apr 14 at 23:15

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