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? 
 A: If I did this correctly:
\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}
A: From https://www.jstor.org/stable/2286081 (Exact Covariance of Products of Random Variables)  provides the general formulae. Assuming multivariate normality it is:
$$ \mathrm{cov}(xy, uv)  = \mathrm{E}(x)\,\mathrm{E}(u)\, \mathrm{cov}(y, v) +  
    \mathrm{E}(x)\,\mathrm{E}(v)\,\mathrm{cov}(y, u) + \\ 
                          \mathrm{E}(y)\,\mathrm{E}(u)\, \mathrm{cov}(x, v) +  
    \mathrm{E}(y)\,\mathrm{E}(v)\,\mathrm{cov}(x, u) + \\
    \mathrm{cov}(x, u)\, \mathrm{cov}(y, v) + \mathrm{cov}(x, v)\,\mathrm{cov}(y, u)
$$
In your case: x = A, y=C, u=B, v =D  and the only non cero covariances are
cov(A,B)= cov(x,u) and cov(C,D)=cov(y,v) so:
$$ \mathrm{cov}(AC, BD)  = \mathrm{E}(A)\,\mathrm{E}(B)\, \mathrm{cov}(C, D) +  \\ 
                           \mathrm{E}(C)\,\mathrm{E}(D)\,\mathrm{cov}(A, B) + \\
    \mathrm{cov}(A, B)\, \mathrm{cov}(C, D) 
$$
Which is the result of the previous post.
