The Problem:

Let $X_1, X_2, X_3$ be random variables with means $\mu_1, \mu_2, \mu_3$, variances $\sigma_1^2, \sigma_2^2, \sigma_3^2$ and correlation coefficients $\rho_{12}, \rho_{13}, \rho_{23}$ Suppose the following:

$$E(X_1 - \mu_1|X_2,X_3)=b_2(x_2-\mu_2)-b_3(x_3-\mu_3)$$

Determine $b_2$ and $b_3$ in terms of the variances and correlation coefficients.

My Work:

I see that on the right hand side of the equation, the makings of $\sigma_2^2$ and $\sigma_3^2$ are multiplied by their respective constants, so I started by isolating the $b_2$ term, squaring both sides, and then taking the expectation of both sides with respect to $X_2$:

$$E_{x_2}([E(X_1 -\mu_1|X_2,X_3)-b_3(x_3-\mu_3)]^2)= b_2^2E([x_2-\mu_2]^2) = b_2^2\sigma_2^2$$

This seems very, very messy so I doubt I am heading in the right direction. Also, I have no idea how to pull the correlation coefficients out.


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