Let's assume i have a random variable $Z_1$ that is the sum of two independent variables $X_1$ and $Y_1$. So $$Z_1 = X_1 + Y_1$$.
I know that $Var(Z_1) = Var(X_1) + Var(Y_1)$, because X and Y are independent. Now I have $Z_2 = X_2 + Y_2$, where $X_2$ and $Y_2$ are not dependent but $X_1$ and $X_2$ and $Y_1$ and $Y_2$ are dependent.
Is it true to say that $Cov(Z_1, Z_2) = Cov(X_1, X_2) + Cov(Y_1, Y_2)$ ?
Thanks!
On
Exploit that covariance is bilinear:
$$\mathsf{Cov}(X_1+Y_1,X_2+Y_2)=\mathsf{Cov}(X_1,X_2)+\mathsf{Cov}(X_1,Y_2)+\mathsf{Cov}(Y_1,X_2)+\mathsf{Cov}(Y_1,Y_2)$$
In a comment you confirm that $X_1$ and $Y_2$ are independent, and that $X_2$ and $Y_1$ are independent, so we can take out the second and third term. What remains is:$$\mathsf{Cov}(X_1,X_2)+\mathsf{Cov}(Y_1,Y_2)$$
Eventual independence of $X_1,Y_1$ and/or $X_2,Y_2$ is irrelevant.
$$Cov[Z_1,Z_2]=\mathbb{E}[Z_1Z_2]-\mathbb{E}[Z_1]\mathbb{E}[Z_2]$$ $$\mathbb{E}[Z_1Z_2]=\mathbb{E}[(X_1+Y_1)(X_2+Y_2)]=\mathbb{E}[X_1X_2+X_1Y_2+Y_1X_2+Y_1Y_2]$$ Since only $X_1,X_2$ and $Y_1,Y_2$ are dependent we end up with $$\mathbb{E}[X_1X_2]+\mathbb{E}[X_1]\mathbb{E}[Y_2]+\mathbb{E}[Y_1]\mathbb{E}[X_2]+\mathbb{E}[Y_1Y_2]$$ Now the second term of the covariance is $$\mathbb{E}[Z_1]\mathbb{E}[Z_2]=\mathbb{E}[(X_1+Y_1)]\mathbb{E}[(X_2+Y_2)]=\mathbb{E}[X_1]\mathbb{E}[X_2]+\mathbb{E}[X_1]\mathbb{E}[Y_2]+\mathbb{E}[Y_1]\mathbb{E}[X_2]+\mathbb{E}[Y_1]\mathbb{E}[Y_2]$$ So in the end we get $$Cov[Z_1,Z_2]=\mathbb{E}[X_1X_2]-\mathbb{E}[X_1]\mathbb{E}[X_2]+\mathbb{E}[Y_1Y_2]-\mathbb{E}[Y_1]\mathbb{E}[Y_2]=Cov[X_1,X_2]+Cov[Y_1,Y_2]$$