Theorem. If $X$ and $Y$ are two independent normal random variables with means $a,b$ and variances $c,d$ respectly, the sum $X+Y$ is a normal random variable with parameters $a+b$ and $c+d$.
My question is, I know that $X$ is normal with parameters $a$ and $c$ if and only if $Z=(X-a)/\surd c$ is standard normal (ie: with parameters $0$ and $1$). If I show the Theorem above for the case with two standard normal random variables $E$ and $F$ say, can I simply conclude that the result is true for the general case by making:
$X=\frac{(E-a)}{\surd c}$ and $Y=\frac{(F-b)}{\surd d}$ and $X+Y= \frac{(E+F)-(a+b)}{\surd c+\surd d} $??
Well, $E=\frac{X-a}{\surd c}$, $F=\frac{Y-b}{\surd d}$ are independent standard normal random variables, but $E+F$ is distributed $\mathcal N(0,2)$ .
$a+b + E\surd c+ F\surd d$ is distributed $\mathcal N(a+b, c+d)$
$E\surd c+F\surd d$ is distributed $\mathcal N(0,c+d)$
$\frac{E\surd c+F\surd d}{\surd(c+d)}$ is distributed $\mathcal N(0,1)$