Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Covariance ▷ Relationship to correlation

Theorem: Let $X$ and $Y$ be random variables. Then, the covariance of $X$ and $Y$ is equal to the product of their correlation and the standard deviations of $X$ and $Y$:

\[\label{eq:cov-corr} \mathrm{Cov}(X,Y) = \sigma_X \, \mathrm{Corr}(X,Y) \, \sigma_Y \; .\]

Proof: The correlation of $X$ and $Y$ is defined as

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

which can be rearranged for the covariance to give

\[\label{eq:cov-corr-qed} \mathrm{Cov}(X,Y) = \sigma_X \, \mathrm{Corr}(X,Y) \, \sigma_Y\]
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Metadata: ID: P119 | shortcut: cov-corr | author: JoramSoch | date: 2020-06-02, 21:00.