I have a question about the conditional expectation of a multivariate normal distribution when the condition is based on an inequality. Suppose $x_{1}=\theta+\varepsilon_{1}$ and $x_{2}=\theta+\varepsilon_{2}$ where $\theta \sim N\left(y,\; \tau^2 \right)$ is a constant and $\varepsilon_{1}\;\varepsilon_{2}$ follow $i.i.d.$ distribution $N\left(0,\;\sigma^2\right)$. In that case, how do we compute $E\left[\theta | x_{1}=k, x_{2}>k \right]$ where $k$ is a constant?
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