Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Correlation ▷ Conditional correlation

Definition: Let $X$, $Y$ and $Z$ be random variables with the joint probability distribution $p(X,Y,Z)$ and consider the conditional probability distribution $p(X,Y \vert Z)$ as well as the (marginal and) conditional probability distributions $p(X \vert Z)$ and $p(Y \vert Z)$. Then, the conditional correlation of $X$ and $Y$ given $Z$ is defined as

\[\label{eq:corr-cond} \mathrm{Corr}(X,Y|Z) = \frac{\mathrm{Cov}(X,Y|Z)}{\sqrt{\mathrm{Var}(X|Z)} \sqrt{\mathrm{Var}(Y|Z)}}\]

where $\mathrm{Cov}(X,Y \vert Z)$ is the conditional covariance of $X$ and $Y$, i.e. the covariance of $p(X,Y \vert Z)$, and $\mathrm{Var}(X \vert Z)$ as well as $\mathrm{Var}(Y \vert Z)$ are the conditional variances of $X$ and $Y$, i.e. the variances of $p(X \vert Z)$ and $p(Y \vert Z)$.

 
Sources:

Metadata: ID: D226 | shortcut: corr-cond | author: JoramSoch | date: 2026-03-26, 10:03.