Index: The Book of Statistical ProofsGeneral 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:

$\label{eq:var-add-qed} \mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) \; .$
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Metadata: ID: P130 | shortcut: var-add | author: JoramSoch | date: 2020-07-07, 06:52.