What is $Var[A+B-2C]$?

Question

I'm aware that $$Var[A+B] = Var[A] + Var[B] + 2Cov[A,B]$$ and $$Var[2C] = 4Var[C].$$ So then $$Var[A+B-2C] = Var[A] + Var[B] + 4Var[C] + 2Cov[A,B] - 4Cov[B,C] - 4Cov[A,C].$$ Is this correct?

Answer 1

It is called the Bilinearity of Covariance.

$$\begin{align}\mathsf{Cov}\left(\raise{0.75ex}{\sum_{i\in\mathcal I} a_iX_i,\sum_{j\in\mathcal J} b_jY_j}\right)&=\sum_{\langle i,j\rangle\in\mathcal {I{\times}J}} a_ib_j~\mathsf{Cov}(X_i,Y_j)\end{align}$$

In the specific case of variance

$$\begin{align}\mathsf{Var}\left(\raise{0.75ex}{\sum_{i\in\mathcal I} a_iX_i}\right)&=\sum_{\langle i,j\rangle\in\mathcal {I^2}} a_ia_j~\mathsf{Cov}(X_i,X_j)\\[1ex]&=\sum_{i\in\mathcal I} a_i^2~\mathsf{Var}(X_i)+2~\mathop{\sum\qquad}_{\langle i,j\rangle\in\mathcal I^2:i<j}~a_i a_j~\mathsf{Cov}(X_i,X_j)\end{align}$$

So indeed$${\mathsf{Var}(A+B-2C)}={{\mathsf{Var}(A)+\mathsf{Var}(B)+4~\mathsf{Var}(C)}+{2~\mathsf{Cov}(A,B)-4~\mathsf{Cov}(A,C)-4~\mathsf{Cov}(B,C)}}$$