added proof "corr-ind"

Author: JoramSoch

P/corr-ind.md

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+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-09-27 12:32:50
+
+title: "Independent random variables are uncorrelated"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Correlation"
+theorem: "Correlation under independence"
+
+sources:
+  - authors: "StatProofBook"
+    year: 2022
+    title: "Uncorrelated random variables are not necessarily independent."
+    in: "X"
+    pages: "Nov 22, 2022, 06:34 AM"
+    url: "https://x.com/StatProofBook/status/1594927275514134528"
+
+proof_id: "P472"
+shortcut: "corr-ind"
+username: "JoramSoch"
+---
+
+
+**Theorem:** [Independent](/D/ind) [random variables](/D/rvar) are uncorrelated.
+
+
+**Proof:** The [correlation](/D/corr) 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](/P/cov-ind):
+
+$$ \label{eq:cov-ind}
+X, Y \; \text{independent} \quad \Rightarrow \quad \mathrm{Cov}(X,Y) = 0 \; .
+$$
+
+Thus, the correlation of independent random variables is also zero:
+
+$$ \label{eq:corr-ind-qed}
+X, Y \; \text{independent} \quad \Rightarrow \quad \mathrm{Corr}(X,Y) = 0 \; .
+$$