Covariance of transformation of correlated Gaussian Random Variables

Question

Suppose $X_1 \sim N(0,\frac{3}{\pi e})$ and $X_2 \sim N(0,\frac{3}{\pi e})$ and $E[X_1X_2]=\frac{1}{\pi e}$. Next, let $V_1 = 3X_1+X_2$ and $V_2 = 2X_1+X_2$. What is Cov$(V_1,V_2)$?

Clearly Cov$(X_1,X_2) = \frac{1}{\pi e}$. Moreover, we have that $V_1 \sim N(0,\frac{30}{\pi e})$ and $V_2 \sim N(0, \frac{15}{\pi e})$. However, I'm not sure how to find the Cov$(V_1,V_2)$.

Answer 1

$\DeclareMathOperator{\Cov}{Cov}$ $\DeclareMathOperator{\Var}{Var}$ Simply use the bilinearity property of the covariance operator and the definition $\Var(X) = \Cov(X, X)$: \begin{align} & \Cov(V_1, V_2) = \Cov(3X_1 + X_2, 2X_1 + X_2) \\ =& 6\Var(X_1) + 3\Cov(X_1, X_2) + 2\Cov(X_1, X_2) + \Var(X_2) \\ =& 6\Var(X_1) + 5\Cov(X_1, X_2) + \Var(X_2) \\ =& 6\frac{3}{\pi e} + 5\frac{1}{\pi e} + \frac{3}{\pi e} = \frac{26}{\pi e}. \end{align}