Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Correlation ▷ Correlation under independence

Theorem: Independent random variables are uncorrelated.

Proof: The correlation of two random variables is defined as:

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

The covariance of independent random variables is zero:

\[\label{eq:cov-ind} X, Y \; \text{independent} \quad \Rightarrow \quad \mathrm{Cov}(X,Y) = 0 \; .\]

Thus, the correlation of independent random variables is also zero:

\[\label{eq:corr-ind-qed} X, Y \; \text{independent} \quad \Rightarrow \quad \mathrm{Corr}(X,Y) = 0 \; .\]
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Metadata: ID: P472 | shortcut: corr-ind | author: JoramSoch | date: 2024-09-27, 12:32.