Proof: Invariance of the covariance under addition of constants
Index:
The Book of Statistical Proofs ▷
General Theorems ▷
Probability theory ▷
Covariance ▷
Invariance under addition
Metadata: ID: P539 | shortcut: cov-inv | author: JoramSoch | date: 2026-05-28, 10:25.
Theorem: The covariance is invariant under addition of constants:
\[\label{eq:cov-inv} \mathrm{Cov}(X+a,Y+b) = \mathrm{Cov}(X,Y) \; .\]Proof: The covariance is defined in terms of the expected value as
\[\label{eq:cov} \mathrm{Cov}(X,Y) = \mathrm{E}\left[ (X-\mathrm{E}(X)) (Y-\mathrm{E}(Y)) \right] \; .\]Using this and the linearity of the expected value, we can derive \eqref{eq:cov-inv} as follows:
\[\label{eq:cov-inv-qed} \begin{split} \mathrm{Cov}(X+a,Y+b) &\overset{\eqref{eq:cov}}{=} \mathrm{E}\left[ ((X+a)-\mathrm{E}(X+a)) ((Y+b)-\mathrm{E}(Y+b)) \right] \\ &= \mathrm{E}\left[ (X + a - \mathrm{E}(X) - a) (Y + b - \mathrm{E}(Y) - b) \right] \\ &= \mathrm{E}\left[ (X-\mathrm{E}(X)) (Y-\mathrm{E}(Y)) \right] \\ &\overset{\eqref{eq:cov}}{=} \mathrm{Cov}(X,Y) \; . \\ \end{split}\]∎
Sources: Metadata: ID: P539 | shortcut: cov-inv | author: JoramSoch | date: 2026-05-28, 10:25.