Let $X,Y$ be independent random variables with normal distributions $N(6,4)$ and $N(6,7)$ respectively. Then:
$(A)$ Random variable $Z:=X+Y$ has density $f(x)=\frac{1}{22\sqrt\pi}e^{-\frac{(x-12)^2}{22}}$ ;
$(B)$ Random variable $Z:=X-Y$ may take negative values ;
$(C)$ Random variable $Z:=X-Y$ only takes negative values ;
I have to prove or disprove these.
$ad. (A)\;\;$ proper answer is $f(x)=\frac{1}{\sqrt\pi\sqrt{22}}e^{-\frac{(x-12)^2}{22}}$ so it's wrong ;
$ad. (B)$ and $(C)\;\;$ if $N(6,4)+N(6,7) = N(12,11)$, then is it right that $N(6,4)-N(6,7) = N(0,-3)$? I suppose that it's not, but then I have no idea how to make this one. Should I do anything with $P(Z\leq t)=P(X-Y \leq t)=...$ ?
Any help will be appreciated.