Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Variance ▷ Variance of linear combination

Theorem: The variance of the linear combination of two random variables is a function of the variances as well as the covariance of those random variables:

\[\label{eq:var-lincomb} \mathrm{Var}(aX+bY) = a^2 \, \mathrm{Var}(X) + b^2 \, \mathrm{Var}(Y) + 2ab \, \mathrm{Cov}(X,Y) \; .\]

Proof: The variance is defined in terms of the expected value as

\[\label{eq:var} \mathrm{Var}(X) = \mathrm{E}\left[ (X-\mathrm{E}(X))^2 \right] \; .\]

Using this and the linearity of the expected value, we can derive \eqref{eq:var-lincomb} as follows:

\[\label{eq:var-lincomb-qed} \begin{split} \mathrm{Var}(aX+bY) &\overset{\eqref{eq:var}}{=} \mathrm{E}\left[ ((aX+bY)-\mathrm{E}(aX+bY))^2 \right] \\ &= \mathrm{E}\left[ (a [X-\mathrm{E}(X)] + b [Y-\mathrm{E}(Y)])^2 \right] \\ &= \mathrm{E}\left[ a^2 \, (X-\mathrm{E}(X))^2 + b^2 \, (Y-\mathrm{E}(Y))^2 + 2ab \, (X-\mathrm{E}(X)) (Y-\mathrm{E}(Y)) \right] \\ &= \mathrm{E}\left[ a^2 \, (X-\mathrm{E}(X))^2 \right] + \mathrm{E}\left[ b^2 \, (Y-\mathrm{E}(Y))^2 \right] + \mathrm{E}\left[ 2ab \, (X-\mathrm{E}(X)) (Y-\mathrm{E}(Y)) \right] \\ &\overset{\eqref{eq:var}}{=} a^2 \, \mathrm{Var}(X) + b^2 \, \mathrm{Var}(Y) + 2ab \, \mathrm{Cov}(X,Y) \; . \\ \end{split}\]
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Metadata: ID: P129 | shortcut: var-lincomb | author: JoramSoch | date: 2020-07-07, 06:21.