Sum of two independent normal distributed random variables

Question

If $X_i$, $i =1,2$ are independent and have normal distribution with mean $0$ and variance $\sigma_i ^2$. Show that $X_1 + X_2$ has a normal distribution with mean $0$ and variance $\sigma_1^2 + \sigma_2^2$.

Any idea is appreciated.

Answer 1

Use moment generating function $$M_{X_1+X_2}(t)=\exp(\sigma_1^2t^2/2+\sigma_2^2t^2/2)=\exp((\sigma_1^2+\sigma_2^2)t^2/2)$$ and so $X_1+X_2\sim N(0,\sigma_1^2+\sigma_2^2)$.