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

Theorem: The covariance scales upon multiplication with constants:

\[\label{eq:cov-scal} \mathrm{Cov}(aX,bY) = a b \, \mathrm{Cov}(X,Y) \; .\]

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-scal} as follows:

\[\label{eq:cov-scal-qed} \begin{split} \mathrm{Cov}(aX,bY) &\overset{\eqref{eq:cov}}{=} \mathrm{E}\left[ ((aX)-\mathrm{E}(aX)) ((bY)-\mathrm{E}(bY)) \right] \\ &= \mathrm{E}\left[ (aX-a\mathrm{E}(X)) (bY-b\mathrm{E}(Y)) \right] \\ &= \mathrm{E}\left[ a (X-\mathrm{E}(X)) b (Y-\mathrm{E}(Y)) \right] \\ &= a b \, \mathrm{E}\left[ (X-\mathrm{E}(X)) (Y-\mathrm{E}(Y)) \right] \\ &\overset{\eqref{eq:cov}}{=} a b \, \mathrm{Cov}(X,Y) \; . \\ \end{split}\]
∎
Sources:

Metadata: ID: P540 | shortcut: cov-scal | author: JoramSoch | date: 2026-05-28, 10:36.