Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Correlation ▷ Invariance under linear transformation

Theorem: Let $X$ and $Y$ be two random variables. Then, the correlation of any linear transformations of $X$ and $Y$ is equal to the correlation of $X$ and $Y$ up to a possible change in sign:

\[\label{eq:corr-inv} \mathrm{Corr}(aX+b,cY+d) = \pm \mathrm{Corr}(X,Y) \; .\]

Proof: The correlation is defined in terms of covariance and variance as

\[\label{eq:corr} \mathrm{Corr}(X,Y) = \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)} \sqrt{\mathrm{Var}(Y)}} \; .\]

The covariance is invariant under addition and scales upon multiplication with constants:

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

Variances are also invariant under addition and scale upon multiplication with constants:

\[\label{eq:var-inv-scal} \begin{split} \mathrm{Var}(aX+b) &= a^2 \mathrm{Var}(X) \\ \mathrm{Var}(cY+d) &= c^2 \mathrm{Var}(Y) \; . \end{split}\]

Taking all this together, we obtain:

\[\label{eq:corr-inv-qed} \begin{split} \mathrm{Corr}(aX+b,cY+d) &\overset{\eqref{eq:corr}}{=} \frac{\mathrm{Cov}(aX+b,cY+d)}{\sqrt{\mathrm{Var}(aX+b)} \sqrt{\mathrm{Var}(cY+d)}} \\ &\overset{\eqref{eq:cov-inv-scal}}{=} \frac{a c \, \mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(aX+b)} \sqrt{\mathrm{Var}(cY+d)}} \\ &\overset{\eqref{eq:var-inv-scal}}{=} \frac{a c \, \mathrm{Cov}(X,Y)}{\sqrt{a^2 \mathrm{Var}(X)} \sqrt{c^2 \mathrm{Var}(Y)}} \\ &= \frac{a c}{\sqrt{a^2} \sqrt{c^2}} \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)} \sqrt{\mathrm{Var}(Y)}} \\ &= \frac{a c}{|a| |c|} \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)} \sqrt{\mathrm{Var}(Y)}} \\ &\overset{\eqref{eq:corr}}{=} \pm \mathrm{Corr}(X,Y) \; . \end{split}\]
∎
Sources:

Metadata: ID: P541 | shortcut: corr-inv | author: JoramSoch | date: 2026-05-28, 16:28.