var-add.md

layout	mathjax	author	affiliation	e_mail	date	title	chapter	section	topic	theorem	sources	proof_id	shortcut	username
proof	true	Joram Soch	BCCN Berlin	joram.soch@bccn-berlin.de	2020-07-06 23:52:00 -0700	Additivity of the variance for independent random variables	General Theorems	Probability theory	Variance	Additivity under independence	authors year title in pages url Wikipedia 2020 Variance Wikipedia, the free encyclopedia retrieved on 2020-07-07 https://en.wikipedia.org/wiki/Variance#Basic_properties	P130	var-add	JoramSoch

Theorem: The variance is additive for independent random variables:

$$ \label{eq:var-add} p(X,Y) = p(X) , p(Y) \quad \Rightarrow \quad \mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) ; . $$

Proof: The variance of the sum of two random variables is given by

$$ \label{eq:var-sum} \mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2 , \mathrm{Cov}(X,Y) ; . $$

The covariance of independent random variables is zero:

$$ \label{eq:cov-ind} p(X,Y) = p(X) , p(Y) \quad \Rightarrow \quad \mathrm{Cov}(X,Y) = 0 ; . $$

Combining \eqref{eq:var-sum} and \eqref{eq:cov-ind}, we have:

$$ \label{eq:var-add-qed} \mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) ; . $$