Self-covariance equals variance

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Covariance ▷ Self-covariance

Theorem: The covariance of a random variable with itself is equal to the variance:

\[\label{eq:cov-var} \mathrm{Cov}(X,X) = \mathrm{Var}(X) \; .\]

Proof: The covariance of random variables $X$ and $Y$ is defined as:

\[\label{eq:cov} \mathrm{Cov}(X,Y) = \mathrm{E}\left[ (X-\mathrm{E}[X]) (Y-\mathrm{E}[Y]) \right] \; .\]

Inserting $X$ for $Y$ in \eqref{eq:cov}, the result is the variance of $X$:

\[\label{eq:cov-var-qed} \begin{split} \mathrm{Cov}(X,X) &\overset{\eqref{eq:cov}}{=} \mathrm{E}\left[ (X-\mathrm{E}[X]) (X-\mathrm{E}[X]) \right] \\ &= \mathrm{E}\left[ (X-\mathrm{E}[X])^2 \right] \\ &= \mathrm{Var}(X) \; . \end{split}\]

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Sources:

Wikipedia (2022): "Covariance"; in: Wikipedia, the free encyclopedia, retrieved on 2022-09-26; URL: https://en.wikipedia.org/wiki/Covariance#Covariance_with_itself.

Metadata: ID: P352 | shortcut: cov-var | author: JoramSoch | date: 2022-09-26, 12:08.