Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Covariance ▷ Symmetry

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:

\[\label{eq:cov-symm-qed} \begin{split} \mathrm{Cov}(Y,X) &\overset{\eqref{eq:cov}}{=} \mathrm{E}\left[ (Y-\mathrm{E}[Y]) (X-\mathrm{E}[X]) \right] \\ &= \mathrm{E}\left[ (X-\mathrm{E}[X]) (Y-\mathrm{E}[Y]) \right] \\ &= \mathrm{Cov}(X,Y) \; . \end{split}\]
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Metadata: ID: P353 | shortcut: cov-symm | author: JoramSoch | date: 2022-09-26, 12:14.