Distribution of the sum of two independent normal variables

Question

Given the two variables $A\sim \text{N}(\mu, \phi^2)$ and $B\sim \text{N}(\xi, \omega^2)$ with $\mu, \xi \in R$ and $\phi^2, \omega^2 > 0$ how do I prove that $C := A + B\sim \text{N}(\mu + \xi, \phi^2 + \omega^2)$ ?

Answer 1

in case they are independent use Characteristic functions: $$\phi_{A+B} (t) = \phi_{A}(t) \phi_{B}(t)$$ and it is easy to see that $\phi_{A}(t)$ looks like char. function. of $N(\mu+\xi, \omega^{2}+\phi^{2})$