Proof: Symmetry of the covariance
Index:
The Book of Statistical Proofs ▷
General Theorems ▷
Probability theory ▷
Covariance ▷
Symmetry
Metadata: ID: P353 | shortcut: cov-symm | author: JoramSoch | date: 2022-09-26, 12:14.
Theorem: The covariance of two random variables is a symmetric function:
\[\label{eq:cov-symm} \mathrm{Cov}(X,Y) = \mathrm{Cov}(Y,X) \; .\]Proof: The covariance of random variables $X$ and $Y$ is defined as:
\[\label{eq:cov} \mathrm{Cov}(X,Y) = \mathrm{E}\left[ (X-\mathrm{E}[X]) (Y-\mathrm{E}[Y]) \right] \; .\]Switching $X$ and $Y$ in \eqref{eq:cov}, we can easily see:
\[\label{eq:cov-symm-qed} \begin{split} \mathrm{Cov}(Y,X) &\overset{\eqref{eq:cov}}{=} \mathrm{E}\left[ (Y-\mathrm{E}[Y]) (X-\mathrm{E}[X]) \right] \\ &= \mathrm{E}\left[ (X-\mathrm{E}[X]) (Y-\mathrm{E}[Y]) \right] \\ &= \mathrm{Cov}(X,Y) \; . \end{split}\]∎
Sources: - Wikipedia (2022): "Covariance"; in: Wikipedia, the free encyclopedia, retrieved on 2022-09-26; URL: https://en.wikipedia.org/wiki/Covariance#Covariance_of_linear_combinations.
Metadata: ID: P353 | shortcut: cov-symm | author: JoramSoch | date: 2022-09-26, 12:14.