Proof: Additivity of the variance for independent random variables

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Variance ▷ Additivity under independence

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.