If two random variables have an equal normal distribution, then how to find a cdf of the sum of those variables?

Question

So if $X \sim \mathcal{N}(\mu_1,\sigma^2_1) \text{ and } Y \sim\mathcal{N}(\mu_2,\sigma^2_2) \text{ then what will be the cdf of } Z=X+Y?$

Answer 1

Assuming that $X$ and $Y$ are independent, from the characteristic function $\varphi_X(t) = e^{i\mu_1 t - \sigma_1^2t^2/2}$ we see that the characteristic function of the sum $X+Y$ is the product of $\varphi_X$ and $\varphi_Y$: $$ \varphi_{X+Y}(t) = e^{i\mu_1 t - \sigma_1^2t^2/2}e^{i\mu_2 t - \sigma_2^2t^2/2} = e^{i(\mu_1+\mu_2t -(\sigma_1^2+\sigma_2^2)t^2/2}, $$ from which it follows that $X+Y\sim\mathrm{N}(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)$.