$A, B, X$ are random variables.
If we know $corr(A, X)=0.03, corr(B, X)=0.04, corr(A, B)=0$
What's the range of the correlation between an arbitrary linear combination of $A, B$ and $Y$ i.e. $corr(c_1A + c_2B, X)$?
I tried to simplify it by using $Cov(X, Y) = E(XY) - E(X)E(Y)$ and rules of $Var()$ calculation, but I got an expression in which the range still seems unclear.
By using bilinerity formula of Cov, it shows that $$\rho = \frac{\rho_{A,X} c_1\sigma_A + \rho_{B,X} c_2\sigma_B}{\sqrt{c_1^2\sigma_A^2 + c_2^2\sigma_B^2}}$$ (derived by using $corr(A, B) = 0$), I can imagine this as to do projection of $(\rho_{AX}, \rho_{BX})$ into the direction of $(c_1, c_2)$ in the $\sigma_A, \sigma_B$ space. Since it could be any direction, the range shall be within range $$(-\sqrt{\rho_{A,X}^2 + \rho_{B,X}^2}, \sqrt{\rho_{A,X}^2 + \rho_{B,X}^2})$$ if we take $corr_a = 0.03, corr_b=0.04$, the range should be $(-0.05, 0.05)$. Not sure if this is the correct answer.