Proof: Invariance of the variance under addition of a constant
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The Book of Statistical Proofs ▷
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Invariance under addition
Metadata: ID: P126 | shortcut: var-inv | author: JoramSoch | date: 2020-07-07, 05:23.
Theorem: The variance is invariant under addition of a constant:
\[\label{eq:var-inv} \mathrm{Var}(X+a) = \mathrm{Var}(X)\]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-inv} as follows:
\[\label{eq:var-inv-qed} \begin{split} \mathrm{Var}(X+a) &\overset{\eqref{eq:var}}{=} \mathrm{E}\left[ ((X+a)-\mathrm{E}(X+a))^2 \right] \\ &= \mathrm{E}\left[ (X + a - \mathrm{E}(X) - a)^2 \right] \\ &= \mathrm{E}\left[ (X-\mathrm{E}(X))^2 \right] \\ &\overset{\eqref{eq:var}}{=} \mathrm{Var}(X) \; . \\ \end{split}\]∎
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: P126 | shortcut: var-inv | author: JoramSoch | date: 2020-07-07, 05:23.