Variance random variables

Question

someone can explain me why the difference of 2 random variables indipendent is the sum of 2 random variables and not the difference? So, why Var(X-Y) = Var(x) + Var(y) ?

Answer 1

Because in general $$\operatorname{Var}(aX+bY)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y)+2ab\operatorname{Cov}(X,Y).$$

Now if the random variables are independent, we get with $a=1$ and $b=-1$: $$\begin{align}\operatorname{Var}(X-Y)&= \operatorname{Var}\bigl(1\cdot X+(-1)\cdot Y\bigr)\\ &=1^2\operatorname{Var}(X)+(-1)^2\operatorname{Var}(Y)+2ab\cdot0\\ &=\operatorname{Var}(X)+\operatorname{Var}(Y).\end{align}$$