Scaling of the variance upon multiplication with a constant

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Variance ▷ Scaling upon multiplication

Theorem: The variance scales upon multiplication with a constant:

\[\label{eq:var-scal} \mathrm{Var}(aX) = a^2 \, \mathrm{Var}(X)\]

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-scal} as follows:

\[\label{eq:var-scal-qed} \begin{split} \mathrm{Var}(aX) &\overset{\eqref{eq:var}}{=} \mathrm{E}\left[ ((aX)-\mathrm{E}(aX))^2 \right] \\ &= \mathrm{E}\left[ (aX - a\mathrm{E}(X))^2 \right] \\ &= \mathrm{E}\left[ (a [X - \mathrm{E}(X)])^2 \right] \\ &= \mathrm{E}\left[ a^2 (X - \mathrm{E}(X))^2 \right] \\ &= a^2 \, \mathrm{E}\left[ (X - \mathrm{E}(X))^2 \right] \\ &\overset{\eqref{eq:var}}{=} a^2 \, \mathrm{Var}(X) \; . \\ \end{split}\]

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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: P127 | shortcut: var-scal | author: JoramSoch | date: 2020-07-07, 05:38.