Probability theory and stochastic processes Theorem

Sum of independent normal random variables

Builds on Normal distribution Characteristic function of the normal distribution Characteristic function of a sum of independent random variables The characteristic function determines the distribution Independence of two 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.

Reading path

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The rationals form an ordered field statement
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Discrete random variable definition
Law of a random variable definition
Independence of two events definition
Random variable with a density definition
Joint probability distribution definition
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Independence of two random variables definition
Mutual independence of events definition
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Marginal distribution definition
Law of the unconscious statistician statement
Independent sigma-algebras definition
Density of the normal distribution definition
Characteristic function definition
Independence means the joint law is the product of the marginals statement
Moment definition
Independence of random variables definition
Normal distribution definition
The characteristic function determines the distribution statement
Expectation of a product of independent random variables statement
Moment-generating function definition
Characteristic function of a sum of independent random variables statement
Moment-generating function of the normal distribution statement
Characteristic function of the normal distribution statement
Sum of independent normal random variables statement

Tag	`0NXGPJWNFN`
URI	`m2://0NXGPJWNFN`
Aliases	`sum-of-normally-distributed-random-variables`
Kind	statement, theorem
Writing	W1 stated
MSC	60E05
Status	This copy of the site has no build data. Run `m2 status 0NXGPJWNFN`.
Uses	Normal distribution, Characteristic function of the normal distribution, Characteristic function of a sum of independent random variables, The characteristic function determines the distribution, Independence of two random variables
Same as	Wikidata Q7636889 Wikipedia Sum of normally distributed random variables

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