# Proof: Relationship between covariance and correlation

**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 $Y$:

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

which can be rearranged for the covariance to give

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

**Sources:**

**Metadata:**ID: P119 | shortcut: cov-corr | author: JoramSoch | date: 2020-06-02, 21:00.