# Covariance of products of dependent random variables

I have four random variables, A, B, C, D, with known mean and variance. As well:

• Cov(A,B) is known and non-zero
• Cov(C,D) is known and non-zero
• A and C are independent
• A and D are independent
• B and C are independent
• B and D are independent

I then create two new random variables:

• X = A*C
• Y = B*D

Is there any way to determine Cov(X,Y) or Var(X+Y)?

If not exactly, is there any way to estimate it? What if I could determine the distributions of A and B and C and D?

$$\begin{eqnarray} \text{Cov}(AC,BD) &=&E(ABCD) - E(AC)E(BD)\\ &=&E(AB)E(CD) - E(A)E(C)E(B)E(D)\\ &=&[E(AB)-E(A)E(B)][E(CD)-E(C)E(D)]+E(A)E(B)[E(CD)-E(C)E(D)]+E(C)E(D)[E(AB)-E(A)E(B)]\\ &=&\text{Cov}(A,B)\text{Cov}(C,D)+E(A)E(B)\text{Cov}(C,D)+E(C)E(D)\text{Cov}(A,B)\end{eqnarray}$$