$X_1, X_2, Y$ are normal random variables,
If we know $\operatorname{corr}(Y, X_1) = \rho_1$, $\operatorname{corr}(Y, X_2) = \rho_2$, $\operatorname{corr}(X_1, X_2) = \rho_3$
What's the range of the correlation between an arbitrary linear combination of $X_1, X_2$ and $Y$ (i.e. $\operatorname{corr}(Y, aX_1 + bX_2)$? where $a, b$ are any real numbers).
If we remove the constraint that $X_1, X_2, Y$ are normal random variables, what are the results?
What I have tried:
I can write
$X_1 = \rho_1 Y + \sqrt{1-\rho_1^2} Z_1$
$X_2 = \rho_2 Y + \sqrt{1-\rho_2^2} Z_2$
But I dont know how to express the relationship of $\operatorname{corr}(X_1, X_2) = \rho_3$ from above ...
Thanks in advance.