Variance on operations of random variables

Question

After reading a bit on the wiki, I though I could find the variance of my variable, but I'm getting confused... Let's assume we have two random variables, $A$ and $B$ which are uncorrelated. Now, what is the variance of:

$D = f*(A-B)+B$

My problem is, from this formula I get:

$Var(D) = f^{2}(Var(A)+Var(B)) + Var(B)$

However, the initial definition of D can be rewritten as:

$D = f*A + (1-f)*B$

which in turns gives

$Var(D) = f^{2}Var(A) + (1-f)^{2}Var(B)$

Maybe I'm seeing wrong, but the two variance formulas are different

Answer 1

If $A$ and $B$ are uncorrelated, $A-B$ and $B$ certainly are correlated. Therefore, you cannot write:

$Var(D) = f^{2}(Var(A)+Var(B)) + Var(B)$

Instead:

$Var(D) = f^{2}(Var(A)+Var(B)) + Var(B) + 2fCovar(A-B,B)$ $Var(D) = f^{2}(Var(A)+Var(B)) + Var(B) - 2fVar(B)$

Which gives, after factorization :

$Var(D) = f^{2}Var(A) + (1-f)^{2}Var(B)$