Correlation between the linear combinations of bivariate normal distributed variables

How can I find the correlation (rho) between $$U$$ and $$V$$, Where $$U = X_1+X_2$$ and $$V= X_1-2X_2$$ $$X_1$$ and $$X_2$$ are normally distributed with $$\mu= 1$$ and $$\sigma= 2$$.

• are $X_1$ and $X_2$ independent? Commented Mar 5, 2020 at 8:13

$$$$\textbf{X}_{p=2} = \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \in N_{p=2}( \bf{ \mu_x }, \bf{ \Sigma_x } ) = N\left( \begin{bmatrix} \mu_1 =1 \\ \mu_2 =1 \end{bmatrix} , \begin{bmatrix} 2 & \sigma_{12} \\ \sigma_{21} & 2 \end{bmatrix} \right)$$$$ Then $$$$\bf{A}\bf{X} = \begin{bmatrix} 1 & 1 \\ 1 & -2 \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} = \begin{bmatrix} X_1 + X_2 \\ X_1 -2X_2 \end{bmatrix} = \begin{bmatrix} U \\ V \end{bmatrix} \in N_{p=2} \left(\bf{A}\bf{\mu_x}, \bf{A}\bf{\Sigma_x}\bf{A}' \right)$$$$ Finally calculate $$\rho = \sigma_{12}/(\sqrt{\sigma_{11}}\sqrt{\sigma_{22}})$$ from the entries of $$\bf{A}\bf{\Sigma}\bf{A}'$$, which is the correlation between U and V. Note that $$\sigma_{11},\sigma_{22}$$ denotes the variances (just notation).
Hint: If $$X_1$$ and $$X_2$$ are independent so $$cov(U,V)=cov(X_1+X_2,X_1-2X_2)$$ $$=cov(X_1,X_1)+cov(X_2,-2X_2)$$ $$=cov(X_1,X_1)-2cov(X_2,X_2)$$