I was wondering if you guys can help me with this one.
Let $X_{1}$ and $X_{2}$ variables such that $X_{1}\sim N(0,1)$ and $X_{2}\sim N(0,1)$. Then, the linear model regression goes as follow, $Y=X\beta$ where $X=[X_{1} X_{2}]$ and $\beta=[\beta_{1},\beta_{2}]^{t}$.
If $r_{12}=Cor(X_{1},X_{2})$ and $r_{jy}=Cor(X_{j},Y)$.
We have that, $$\beta=\frac{1}{1-r_{12}^{2}} \begin{pmatrix} 1&-r_{12}\\ -r_{12}&1 \end{pmatrix} \begin{pmatrix} r_{1y}\\ r_{2y} \end{pmatrix}$$
How can we prove this?!
The model is $y = \beta_1 X_{1} + \beta_2 X_{2}$. Recall that $$ corr(X,Y)= \frac{\mathbb{E}(X-\mathbb{E}X)(Y-\mathbb{E}Y)}{\sigma_X^2 \sigma_Y^2}, $$ in our case $$ corr(X,Y)= \frac{\mathbb{E}(X_1Y)}{1^2 \times 1^2}= \mathbb{E}(X_1Y) = r_{1y}. $$ Now, note that $$ \beta = ( \mathbb{E}(XX^T) )^{-1}\mathbb{E}(X^Ty) $$ where, $$ \mathbb{E}(X ^ T y) = [\mathbb{E}(X_1y), \mathbb{E}(X_2y)] ^ T = [r_{1y}, r_{2y} ]^T. $$ and $$ ( \mathbb{E}(XX^T) )^{-1} = \begin{pmatrix} \mathbb{E}(X_1^2) & \mathbb{E}(X_1 X_2) \\ \mathbb{E}(X_2 X_1) & \mathbb{E}(X_2 ^ 2) \\ \end{pmatrix}^{-1} = \begin{pmatrix} 1 & r_{12} \\ r_{21} & 1 \\ \end{pmatrix}^{-1} = \frac{1}{1 - r_{21}^2}\begin{pmatrix} 1 & - r_{12} \\ - r_{21} & 1 \\ \end{pmatrix}. $$ Properties used: $\mathbb{E} ( X_i^2 ) = \operatorname{var} (X_i) = 1$, $i=1,2$, and $\mathbb{E}(X_1 X_2) = r_{12}$ where $r_{12} = r_{21}$.