Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Variance ▷ Invariance under addition

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:

\[\label{eq:var-inv-qed} \begin{split} \mathrm{Var}(X+a) &\overset{\eqref{eq:var}}{=} \mathrm{E}\left[ ((X+a)-\mathrm{E}(X+a))^2 \right] \\ &= \mathrm{E}\left[ (X + a - \mathrm{E}(X) - a)^2 \right] \\ &= \mathrm{E}\left[ (X-\mathrm{E}(X))^2 \right] \\ &\overset{\eqref{eq:var}}{=} \mathrm{Var}(X) \; . \\ \end{split}\]
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Metadata: ID: P126 | shortcut: var-inv | author: JoramSoch | date: 2020-07-07, 05:23.