Proof: Additivity of the variance for independent random variables
Index:
The Book of Statistical Proofs ▷
General Theorems ▷
Probability theory ▷
Variance ▷
Additivity under independence
Metadata: ID: P130 | shortcut: var-add | author: JoramSoch | date: 2020-07-07, 06:52.
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:
\[\label{eq:var-add-qed} \mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) \; .\]∎
Sources: - Wikipedia (2020): "Variance"; in: Wikipedia, the free encyclopedia, retrieved on 2020-07-07; URL: https://en.wikipedia.org/wiki/Variance#Basic_properties.
Metadata: ID: P130 | shortcut: var-add | author: JoramSoch | date: 2020-07-07, 06:52.