Sum of independent normal random variables

Statement

Let $X \sim \mathcal{N}(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal{N}(\mu_2, \sigma_2^2)$ be independent, with $\sigma_1^2, \sigma_2^2 > 0$. Then $$X + Y \sim \mathcal{N}(\mu_1 + \mu_2, \, \sigma_1^2 + \sigma_2^2).$$

Remarks

Proof with characteristic functions: $\varphi_{X+Y}(t) = \varphi_X(t) \varphi_Y(t) = \exp(i(\mu_1 + \mu_2)t - (\sigma_1^2 + \sigma_2^2) t^2 / 2)$. By the uniqueness theorem, this fixes the law. By induction, a sum of $n$ independent normal variables is normal. Its mean is the sum of the means and its variance is the sum of the variances. With the rule for $aX + b$, every linear combination $a_1 X_1 + \dots + a_n X_n$ of independent normal variables is normal, if some $a_i \ne 0$. The independence condition is necessary. Two normal variables that are not jointly normal can have a sum that is not normal.

Explanations

L1 human

Adding independent normal errors gives another normal error. The means add, and the variances add, but the standard deviations do not add.