# Variance of a Sum of Random Variables with Covariance

Suppose we have the sum of the following random variables,

$$X_T = X_1 - 3X_2 - X_3$$

and we are told that $X_2$ and $X_3$ have a correlation of $-0.2$.

If we wanted to compute the variance of $X_T$ we would consider,

\begin{align} \text{Var}(X_T) &= \text{Var}(X_1 - 3X_2 - X_3)\\ &= \text{Var}(X_1) + 9\text{Var}(X_2) + \text{Var}(X_3) + 2\text{Cov}(...) \end{align}

My question is, would the covariance be,

$$2\text{Cov}(-3X_3,-X_4)$$

Which we can then taken to be,

$$[2\cdot (-3) \cdot (-1)] \text{Cov}(X_3,X_4)$$

• $\mathrm{Var}(\sum a_iX_i)=\sum a_i^2\mathrm{Var}(X_i)+\sum\sum a_ia_j\mathrm{Cov}(X_i,X_j)\quad(i\ne j)$. – StubbornAtom Jun 1 '17 at 8:36