Covariance for stochastic variables

Question

if $X$ and $Y$ are stochastic variables with $\operatorname{Var}(X)=1.34$ and $\operatorname{Cov}(X,Y) = 0.64$, find $\operatorname{Cov}(2X, 3X+2Y)$. No ideas on this one, as I don't see any way of combining the formulas I know to figure this out. I would greatly appreciate some hints

Answer 1

$$\int_{\Omega} (2X - 2\mu_x) (3X - 3\mu_x + 2Y - 2\mu_y) d P = 6 \int_{\Omega} (X - \mu_x) (X - \mu_x) d P + 4 \int_{\Omega} (X - \mu_x) (Y - \mu_y) d P = 6 Var(X) + 4 Cov(X,Y),$$ where $\mu_x := \mathbb E(X)$, $\mu_y := \mathbb E(Y)$, and using the linearity of $\mathbb E$.

Answer 2

\begin{align} \operatorname{Cov}(2X, 3X+2Y) & = 2\operatorname{Cov}(X,3X+2Y) \\[10pt] & = 2\big(\operatorname{Cov}(X,3X)+\operatorname{Cov}(X,2Y) \big) \\[10pt] & = 2\big(3\operatorname{Cov}(X,X) + 2\operatorname{Cov}(X,Y)\big) \\[10pt] & = 6\operatorname{Var}(X) + 4\operatorname{Cov}(X,Y). \end{align}

Answer 3

Hint: Use bilinearity of covariance!