---
tag: 0NXGPJWNFN
kind: statement
role: theorem
name: Sum of independent normal random variables
aliases: [sum-of-normally-distributed-random-variables]
msc: ["60E05"]
uses: [07RXBVTVZB, 0NS0TDA4E2, 02JGYG0T7M, 0T8BYAFEXX, 0E9502NT92]
sources:
  - url: https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
    locator: Independent random variables
same_as:
  wikidata: Q7636889
  wikipedia: Sum of normally distributed 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|0E9502NT92]], 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|0NS0TDA4E2]]: $\varphi_{X+Y}(t) = \varphi_X(t) \varphi_Y(t) = \exp(i(\mu_1 + \mu_2)t - (\sigma_1^2 + \sigma_2^2) t^2 / 2)$. By the [[uniqueness theorem|0T8BYAFEXX]], 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.