Conditional Distribuiton

by Jackson Maike   Last Updated September 12, 2019 02:19 AM

When trying to solve the following question I have to take into account the dependency of $\alpha_2$ to $b_1$, $b_2$ and $S$? Since, apparently, $\alpha_1$ only depends on $\theta_1$ e $\theta_2$. Any suggestions for resolution? Thanks in advance.

Let $$S \sim N(\mu, \Sigma)$$ $$b_i \sim Bern(1 - \omega)$$ $$\theta_i \sim U(-20,40)$$ Regard $\Sigma_{ii} = 1$ and $\Sigma_{ij} = \rho, i \neq j$.

For $$\alpha_1 = 0.7\theta_1 + 0.7\theta_2$$ $$\alpha_1 = 0.7\theta_1 - 0.7\theta_2$$

Find the conditional distribution of $$\alpha_1 | \alpha_2, b_1, b_2, S$$

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