I am trying to prove two statistical formulas and I am having difficulties..
First, I am trying to prove the following:
If we have the random variable: $Y = \sum_{i=1}^n a_i X_i$
Prove that the variance of $Y$ is:
$$\operatorname{Var}(Y)=\sum_{i=1}^n a_i \operatorname{cov}(X_i,Y)$$
I later would like to do the same thing with $W$:
If we have the random variable: $$W = \sum_{i=1}^n b_i Y_i-c_i Z_i$$
Prove that the variance of W is:
$$\operatorname{Var}(W)=\sum_{i=1}^n b_i\operatorname{cov}(Y_i,W)-c_i\operatorname{cov}(Z_i,W)$$
Thank you!